CUMULATIVE VOLUME AND MASS PROFILES FOR DOMINANT STEMS AND
WHOLE TREES
by
Neil R. Ver Planck

A THESIS
Submitted
to Michigan State University
in partial fulﬁllment of the requirements
for the degree of
Forestry – Master of Science
2013

ABSTRACT
CUMULATIVE VOLUME AND MASS PROFILES FOR DOMINANT STEMS AND
WHOLE TREES
by
Neil R. Ver Planck
Integrated whole-tree biomass and volume equations are in great demand due to the simultaneous need to improve estimation of forest carbon stocks and to quantify the distribution of
wood volume within trees for estimating whole-tree utilization potential. While the volume
of the dominant stem in a tree has been extensively studied, the relative mass and volume
of branches has received much less attention. It is particularly challenging to quantify the
branch volume and branch mass in trees with a deliquescent branching architecture (i.e.,
hardwoods) and it is even diﬃcult to model the dominant stem for such trees because the
lack of apical dominance makes deﬁnition of the dominant stem more obscure.
New proﬁle models allow for volume and mass estimation of the dominant stem and
whole tree from ground to the top of the tree. Cumulative branch volume and mass can
be estimated at diﬀerent relative heights from the whole-tree and dominant stem proﬁles
by simple subtraction. The models were developed from intensive, destructive sampling of
32 trees from a temperate hardwood forest in Michigan, USA. The species in the sample
were primarily American beech (Fagus grandifolia Ehrh.) and sugar maple (Acer saccharum
Marsh.). A mixed-eﬀects modeling framework was used throughout model development of
volume, density, and mass proﬁles. Non-linear mixed-eﬀects composite models were used for
cumulative volume proﬁles to account for among tree variation and the ability to correlate
multiple measurements within an individual tree. Species-speciﬁc linear mixed-eﬀects models
were developed for basic density proﬁles. Species-speciﬁc non-linear mixed-eﬀects models
were also developed for cumulative mass proﬁles compared to density-integral and constant
density models.

This thesis is dedicated to Liisa.

iii

ACKNOWLEDGEMENTS

I would like to sincerely thank my thesis advisor and mentor Dr. David W. MacFarlane
for all of his help and guidance throughout my graduate studies. I would also like to thank
my committee members, Dr. Andrew O. Finley and Dr. Bert M. Cregg, for their time and
assistance with this thesis.
Many thanks to the MSU Sustainability Seed Grant and USDA Forest Service’s Forest
Inventory and Analysis program for providing me with funding for my research. A special
thanks to the staﬀ at the Kellogg Experimental Forest: Greg Kowalewski, Mickey Trimner,
Brett Kuipers, and Cody Johnson for assistance in data collection.
Most importantly, a special thanks to my wife, Liisa, for all of her patience, love, and support. I would also like to thank my entire family for all of their support and encouragement
through my graduate study years.

iv

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . .

1

CHAPTER 2 METHODS . . . . . . . . . . . . . . . . .
2.1 Dominant Stem Volume . . . . . . . . . . . . . . . . .
2.2 Whole-tree Volume — Random Branch Sampling . . .
2.3 Density Sampling . . . . . . . . . . . . . . . . . . . . .
2.3.1 Tree Density Sampling . . . . . . . . . . . . . .
2.3.2 Bark Density Sampling . . . . . . . . . . . . . .
2.4 Mass Sampling . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Dominant Stem Mass Sampling . . . . . . . . .
2.4.2 Whole-tree Mass — Random Branch Sampling .
2.5 Study Site . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Tree Selection . . . . . . . . . . . . . . . . . . . . . . .

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16

CHAPTER 3 CUMULATIVE VOLUME PROFILES FOR DOMINANT
STEMS AND WHOLE TREES . . . . . . . . . . . . . . . . . 19
3.1 Related Dominant Stem Taper and Volume Modeling . . . . . . . . . . . . . 19
3.1.1 Dominant Stem Taper Modeling . . . . . . . . . . . . . . . . . . . . . 19
3.1.2 Volume Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1 Dominant Stem Cumulative Volume Models . . . . . . . . . . . . . . 23
3.2.2 Whole-tree Cumulative Volume Models . . . . . . . . . . . . . . . . . 24
3.2.3 Centroid of Dominant Stem and Whole-tree Volume Models . . . . . 25
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.1 Total Volume Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.2 Cumulative Volume Proﬁles . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.3 Centroid of Dominant Stem and Whole-tree Volume Proﬁles . . . . . 34
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
CHAPTER 4 CUMULATIVE MASS AND DENSITY PROFILES FOR
DOMINANT STEMS AND WHOLE TREES . . . . . . . . . 44
4.1 Related Biomass Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.1.1 Diameter at Breast Height Based Biomass Modeling . . . . . . . . . . 44
4.1.2 Component Ratio Method . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.3 Weight-ratio and Density-integral Biomass Modeling . . . . . . . . . 45
4.2 Related Tree Density Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

v

4.4

4.5

4.3.1 Dominant Stem Vertical Density Models . .
4.3.2 Random Branch Path Density Models . . .
4.3.3 Bark Density Models . . . . . . . . . . . . .
4.3.4 Dominant Stem Cumulative Mass Models .
4.3.5 Whole-tree Cumulative Mass Models . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Dominant Stem Vertical Density Proﬁles . .
4.4.2 Random Branch Path Density Proﬁles . . .
4.4.3 Comparative Dominant and Random Branch
4.4.4 Bark Vertical Density Proﬁles . . . . . . . .
4.4.5 Total Mass Estimates . . . . . . . . . . . . .
4.4.6 Cumulative Mass Proﬁles . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Density . . . . . . . . . . . . . . . . . . . .
4.5.2 Mass . . . . . . . . . . . . . . . . . . . . . .

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Path Density Proﬁles
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CHAPTER 5 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

vi

LIST OF TABLES

Table 2.1

Table 2.2

Table 2.3

Table 3.1

Table 3.2

Table 3.3

Table 3.4

Table 3.5

Table 3.6

Table 3.7

Summary of the mean number of trees per ha by species and 10 cm
diameter at breast height (DBH) class for the residual stand. . . . . . . .

14

2

Summary of the average basal area in m by species and 10 cm diameter
ha
at breast height (DBH) class for the residual stand. . . . . . . . . . . . .

15

Species, code, sample size (n), diameter at breast height (DBH), and
height (H) summary statistics for the data set. . . . . . . . . . . . . . . .

18

Species code (spp), number of trees (n), mean dominant stem volume
(VDOM ), mean total volume (VW T ) with standard deviation (SD), minimum (min), and maximum (max) values by species. . . . . . . . . . . . .

26

Species code (spp), diameter at breast height (DBH), DBH class (Dclass),
total dominant stem volume (VDOM ), total volume estimate from the
ﬁrst random branch sample (VW T 1 ), total volume estimate from the
second random branch sample (VW T 2 ), mean total volume (VW T ), standard deviation of total volume (SD), and coeﬃcient of variation of total
volume (CV) summary statistics for each individual tree. . . . . . . . . .

27

Parameter estimates of model 3.8 for the cumulative volume of the dominant stem for the all-species (ALL), American beech (FG), and sugar
maple (AS) mixed-eﬀects models. For the ﬁxed parameters, the standard
error of estimates is given in parantheses. . . . . . . . . . . . . . . . . . .

29

Parameter estimates of model 3.9 for the cumulative whole-tree volume
above relative crown height for the all-species (ALL), American beech
(FG), and sugar maple (AS) mixed-eﬀects models. For the ﬁxed parameters, the standard error of estimates is given in parantheses. . . . . . . .

31

Parameter estimates of model 3.10 for the cumulative whole-tree volume
above relative crown height for the all-species (ALL), American beech
(FG), and sugar maple (AS) mixed-eﬀects models. For the ﬁxed parameters, the standard error of estimates is given in parantheses. . . . . . . .

31

The mean centroid of dominant stem volume by species and 10 cm diameter at breast height (DBH) classes. Standard deviation is given in
parentheses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

The mean centroid of dominant stem volume by species and crown ratio
class (CRC). Standard deviation is given in parentheses. . . . . . . . . . .

36

vii

Table 3.8

Table 3.9

Table 4.1

Table 4.2

Table 4.3

Table 4.4

Table 4.5

Table 4.6

Table 4.7

Table 4.8

Table 4.9

The mean centroid of whole-tree volume classiﬁed by species and 10 cm
diameter at breast height (DBH) classes with the standard deviation
given in parentheses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

The mean centroid of whole-tree volume by species and crown ratio class
(CRC). Standard deviation is given in parentheses. . . . . . . . . . . . . .

39

Relative crown height of the dominant stem (RCH) summary statistics
for all species (ALL), American beech (FG), and sugar maple (AS). . . .

51

Relative height at which the random path diverges from the dominant
stem (RHD) summary statistics for all species (ALL), American beech
(FG), and sugar maple (AS). . . . . . . . . . . . . . . . . . . . . . . . . .

52

The bark sample size by species, dominant (DOM), and random branch
paths (RBP). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

Parameter estimates of models 4.12 - 4.14 for the vertical proﬁle of the
basic density of the dominant stem for the all-species (ALL), American
beech (FG), and sugar maple (AS) mixed-eﬀects models. For the ﬁxed
parameters, the standard error of estimates is given in parantheses. . . . .

58

Parameter estimates of models 4.15 - 4.17 for the basic density proﬁle of
the random branch path for the all-species (ALL), American beech (FG),
and sugar maple (AS) mixed-eﬀects models. For the ﬁxed parameters,
the standard error of estimates is given in parantheses. . . . . . . . . . . .

61

Parameter estimates of models 4.18 - 4.20 for the vertical proﬁle of the
bark basic density of the combined dominant and random branch path
for the all-species (ALL), American beech (FG) and sugar maple (AS)
mixed-eﬀects models. For the ﬁxed parameters, the standard error of
estimates is given in parantheses. . . . . . . . . . . . . . . . . . . . . . . .

66

Species code (spp), number of trees (n), mean dominant stem mass
(MDOM ), mean total mass (MW T ) with standard deviation (SD), minimum (min), and maximum (max) values by species. . . . . . . . . . . . .

70

Parameter estimates of model 4.21 for the cumulative mass of the dominant stem for the all-species (ALL), American beech (FG), and sugar
maple (AS) mixed-eﬀects models. For the ﬁxed parameters, the standard
error of estimates is given in parantheses. . . . . . . . . . . . . . . . . . .

72

Model comparison of the residuals among the cumulative dominant stem
mass proﬁles: (1) non-linear mixed-eﬀects (nlme) model 4.21, (2) densityintegral (integ) model 4.22, and (3) the constant density (const) model
4.23 for all (ALL), American beech (FG), and sugar maple (AS) trees. . .

76

viii

Table 4.10 Parameter estimates of model 4.24 for cumulative whole-tree mass above
relative crown height for the all-species (ALL), American beech (FG),
and sugar maple (AS) mixed-eﬀects models. For the ﬁxed parameters,
the standard error of estimates is given in parantheses. . . . . . . . . . . .

78

Table 4.11 Parameter estimates of model 4.25 for cumulative whole-tree mass above
relative crown height for the all-species (ALL), American beech (FG),
and sugar maple (AS) mixed-eﬀects models. For the ﬁxed parameters,
the standard error of estimates is given in parantheses. . . . . . . . . . . .

78

Table 4.12 Model comparison of the residuals among the cumulative whole-tree mass
proﬁles above relative crown height: (1) non-linear mixed-eﬀects (nlme)
model 4.25, (2) density-integral (integ) model 4.27, and (3) the constant
density (const) model 4.28 for all (ALL), American beech (FG), and
sugar maple (AS) trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

ix

LIST OF FIGURES

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 3.1

A simpliﬁed hardwood tree schematic where the dominant stem is indicated by horizontal lines, the ﬁrst random branch path is indicated
by a checkered pattern, the second random branch path is indicated by
dots, and non-sampled branches are solid. The ﬁrst branch junction
at crown height (CH) demonstrates the length (z) and circumference
measurements taken at each branch junction for the before fork (BF)
and the after fork (AF) locations. . . . . . . . . . . . . . . . . . . . . . .

5

A simpliﬁed hardwood tree schematic showing the location of branch
volume or mass accumulation along the dominant stem as a function of
relative height, ranging from zero at base to one at top of the tree. The
ﬁrst branches encountered are at relative crown height (RCH). . . . . . .

7

The drying time for discs sampled from an American beech tree with a
diameter at breast height, DBH, of 75.4 cm. (a) The oven-drying time
for large discs with diameter outside bark, dob, greater than 50.0 cm.
(b) The oven-drying time for medium discs with dob ranging from 20.1
to 50.0 cm. (c) The oven-drying time for small discs with dob ranging
from 5.1 to 20.0 cm. (d) The oven-drying time for very small discs with
dob ranging from 0.1 to 5.0 cm. Each line and symbol represents a
diﬀerent sampled disc in the respective dob size class. . . . . . . . . . . .

9

(A) Shows the location of the 20.8-hectare research area in the 36hectare second-growth maple-beech stand, ﬁxed-area plots, and sampled
trees within the Fred Russ Experimental Forest. (B) The star represents
the location of the entrance to the Fred Russ Experimental Forest in
Cass County, Michigan. . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Histogram of the relative frequency of the residual stand, represented
by diagonal line ﬁlled bars, and the sampled trees, represented by gray
ﬁlled bars, divided into 10 cm diameter at breast height (DBH) classes. .

17

The total volume, in m3 , of sampled trees for the dominant stem (DOM)
and whole tree (WT) versus diameter at breast height (DBH), in cm,
for all trees (ALL), American beech (FG), and sugar maple (AS). . . . .

28

x

Figure 3.2

Figure 3.3

Figure 3.4

Figure 4.1

Cumulative volume, m3 , proﬁles for the dominant stem (model 3.8),
represented by a dashed line above relative crown height, and whole
tree (model 3.11), represented as the solid line, as a function of relative
height along the dominant stem for the (a) minimum, (b) median, and
(c) maximum diameter at breast height (DBH) American beech (FG)
trees. The vertical dashed line represents relative crown height. . . . . .

32

Cumulative volume, m3 , proﬁles for the dominant stem (model 3.8),
represented by a dashed line above relative crown height, and whole
tree (model 3.11), represented as the solid line, as a function of relative
height along the dominant stem for the (a) minimum, (b) median, and
(c) maximum diameter at breast height (DBH) sugar maple (AS) trees.
The vertical dashed line represents relative crown height. . . . . . . . . .

33

Dominant stem (model 3.8), represented by a dashed line, and wholetree (model 3.11), represented as the solid line, relative volume proﬁles
as a function of relative height along the dominant stem for the (a) minimum, (b) median, and (c) maximum diameter at breast height (DBH)
American beech (FG) trees in the dataset. The horizontal dashed line
indicates a relative volume of 0.5, the centroid of volume. The dotted
vertical line indicates the relative height of the centroid of volume for
the dominant stem (DOM.c), the vertical dashed line represents relative
crown height, and the dash and dot vertical line indicates the relative
height of the centroid of volume for the whole tree (WH.c). . . . . . . . .

40

The mixed-eﬀects of the dominant stem vertical density proﬁle for each
species according to models 4.12 - 4.14. For the median sized American
beech (FG) tree with D = 42.9 cm, and the median sized sugar maple
(AS) tree with D = 42.8 cm and H = 34.0 cm (models 4.13 and 4.14,
respectively). American basswood (TA), black cherry (PS), pignut hickory (CG), and slippery elm (UR) were all ﬁt using the all-species model
(model 4.12) as there was only one individual of each in the sample.
The measured tree densities are shown as triangles. The horizontal dotted line represents the wood density for each species as reported by the
Forest Products Laboratory in Madison, WI (U. S. D. A. Forest Service
2010). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

xi

Figure 4.2

Figure 4.3

Figure 4.4

The mixed-eﬀects vertical density proﬁle for the dominant stem of each
species according to models 4.12 - 4.14, shown as solid lines, versus
relative height. The thicker dashed lines are the mixed-eﬀects density
proﬁle for the random branch path of each species according to models
4.15 - 4.17 versus relative path length. For the median sized American
beech (FG) tree with D = 42.9 cm, and the median sized sugar maple
(AS) tree with D = 42.8 cm and H = 34.0 cm (models 4.13, 4.16 and
4.14, 4.17, respectively). American basswood (TA), black cherry (PS),
pignut hickory (CG), and slippery elm (UR) were all ﬁt using the allspecies models 4.12 and 4.15 as there was only one individual of each in
the sample. The horizontal dotted line represents the wood density for
each species as given by the Forest Products Laboratory in Madison,
WI (U. S. D. A. Forest Service 2010). . . . . . . . . . . . . . . . . . . . .

63

The mixed eﬀects of the vertical density proﬁle for the dominant stem
(solid line) and the random branch path proﬁle (thicker dashed line) for
three American beech trees versus relative height or path length. (a)
Shows an American beech tree with diameter at breast height (DBH)
of 50.1 cm, relative height at which the random path diverges from the
dominant stem (RHD) equal to 0.185. (b) Shows an American beech
tree with DBH of 47.1 cm and RHD equal to 0.544. (c) Shows an
American beech tree with DBH of 42.9 cm and RHD equal to 0.888.
The horizontal dotted line represents the wood density for American
beech of 560 kg/m3 as reported by the Forest Products Laboratory in
Madison, WI (U. S. D. A. Forest Service 2010). . . . . . . . . . . . . . .

64

The mixed eﬀects of the vertical bark density proﬁle for the combined
dominant stem and random branch paths of each species according to
models 4.18 - 4.20, shown as solid lines. The thicker dashed line represents the extrapolation of the models beyond the last measured relative
height. For the median sized American beech (FG) tree with D = 50.1
cm, and the median sized sugar maple (AS) tree with D = 48.1 cm
(models 4.19 and 4.20, respectively). American basswood (TA), black
cherry (PS), pignut hickory (CG), and slippery elm (UR) were all ﬁt
using the all-species model 4.18 as there was only one individual of each
in the sample. The horizontal dotted line represents the bark density
for each species as compiled by the USDA Forest Service (Miles and
Smith 2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

xii

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

The mixed eﬀects of the vertical bark density proﬁle for the combined
dominant stem and random branch paths of each species according to
models 4.18 - 4.20, shown as solid lines, with dashed lines representing the extrapolation of the models beyond the last measured relative
height. For the median sized American beech (FG) tree with D = 50.1
cm, and the median sized sugar maple (AS) tree with D = 48.1 cm
(models 4.19 and 4.20, respectively). American basswood (TA), black
cherry (PS), pignut hickory (CG), and slippery elm (UR) were all ﬁt using the all-species model 4.18 as there was only one individual of each in
the sample. The dotted lines represent the tree density for each species
from models 4.12 - 4.14. . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

The total mass, in kg, of sampled trees for the dominant stem (DOM)
and whole tree (WT) versus diameter at breast height (DBH), in cm,
for all trees (ALL), American beech (FG), and sugar maple (AS). . . . .

71

Cumulative mass, in kg, proﬁles for the dominant stem modeled by:
(1) the non-linear mixed-eﬀects approach (model 4.21, nlme), (2) the
density-integral approach (model 4.22, integ), and (3) the constant density approach (model 4.23, const) as a function of relative height along
the dominant stem for the (a) minimum, (b) median, and (c) maximum diameter at breast height (DBH) American beech (FG) trees in
the dataset. The vertical dashed line represents relative crown height. . .

74

Cumulative mass, in kg, proﬁles for the dominant stem modeled by:
(1) the non-linear mixed-eﬀects approach(model 4.21, nlme), (2) the
density-integral approach (model 4.22, integ), and (3) the constant density approach (model 4.23, const) as a function of relative height along
the dominant stem for the (a) minimum, (b) median, and (c) maximum
diameter at breast height (DBH) sugar maple (AS) trees in the dataset.
The vertical dashed line represents relative crown height. . . . . . . . . .

75

Cumulative mass, in kg, proﬁles for the dominant stem (model 4.21),
represented by a solid line and open triangles above relative crown
height, and whole tree (model 4.26, nlme), represented by a solid line
with ﬁlled-in triangles, as a function of relative height along the dominant stem for the (a) minimum, (b) median, and (c) maximum diameter at breast height (DBH) American beech (FG) trees in the dataset.
Above relative crown height the density-integral (model 4.27, integ) and
constant density (model 4.28, const) models are shown with dotted and
dashed lines, respectively. The vertical dashed line represents relative
crown height. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

xiii

Figure 4.10 Cumulative mass, in kg, proﬁles for the dominant stem (model 4.21),
represented by a solid line and open triangles above relative crown
height, and whole tree (model 4.26, nlme), represented by a the solid
line with ﬁlled-in triangles, as a function of relative height along the
dominant stem for the (a) minimum, (b) median, and (c) maximum
diameter at breast height (DBH) sugar maple (AS) trees in the dataset.
Above relative crown height the density-integral (model 4.27, integ) and
constant density (model 4.28, const) models are shown with dotted and
dashed lines, respectively. The vertical dashed line represents relative
crown height. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiv

81

CHAPTER 1
INTRODUCTION

There is signiﬁcant interest in carbon and the potential role of trees in assimilating CO2 ,
which could potentially mitigate the amount of greenhouse gases polluting the atmosphere.
According to a 2011 report by the United States Environmental Protection Agency (EPA
2011), forested land in the contiguous U.S. sequestered 235.4 million metric tons of C of the
276.8 million metric tons of C sequestered by land-use, land-use change, and forestry. The
later estimate is an oﬀset of 15.3% for total U.S. CO2 emissions. The protocols used in the
estimation follow those outlined by the Intergovernmental Panel on Climate Change 2006
Guidelines for National Greenhouse Gas Inventories (IPCC 2006). Carbon ﬂux estimates for
the forested land category are based on data collected by the U.S. Department of Agriculture
(USDA) Forest Service Inventory and Analysis program (FIA). Carbon is approximately 50
percent of the oven-dry mass of wood with variation around this simpliﬁed estimate being
species-dependent (Lamlom and Savidge 2003). However, estimates may be less certain
for some forest carbon pools in national scale inventories (Woodbury et al. 2007), such as
branches. This research addresses the later by measuring and modeling branches in the
framework of whole-tree volume and mass proﬁles.
Integrated whole-tree mass and volume equations are in great demand due to the simultaneous need to improve estimation of forest carbon stocks and to quantify the distribution
of wood volume within trees for estimating whole-tree utilization potential. While the volume of the dominant stem of a tree has been extensively studied (Hahn and Hansen 1991,
Fonweban et al. 2012), the relative mass and volume of branches has received much less
attention (MacFarlane 2011). It is particularly challenging to quantify the branch volume
and branch mass in trees with a deliquescent branching architecture (i.e., hardwoods) and it
is even diﬃcult to model the dominant stem for such trees because the lack of apical dom-

1

inance makes deﬁnition of the dominant stem more obscure. The focal population of this
research is exclusively northern hardwood trees because of their large and complex branching architecture (Oliver and Larson 1996). This research develops models for the dominant
stem and whole-tree volume, density, and mass. Whole tree in this research is deﬁned as the
aboveground portion of dominant stem and branches outside bark of a tree.
Volume has long been estimated by standard mensuration techniques for the dominant
stem of trees (Avery and Burkhart 2002). The diameter at breast height (DBH, measured
at 1.37 m above ground level) and total tree height are the two most common predictor
variables for total dominant stem and whole-tree volume estimation (Hahn and Hansen
1991). This research expands total volume estimates into proﬁles of cumulative volume at
diﬀerent relative heights. Gregoire and Schabenberger (1996a;b), Fonweban et al. (2012) both
examined cumulative volume proﬁles for the dominant stem only and not the whole tree. In
Chapter 3, the dominant stem and whole-tree cumulative volume models are developed and
discussed.
Mass estimates are generally measured by either direct or indirect methods. The direct
method is to physically weigh all or portions of a tree (Montagu et al. 2005), and the indirect
method converts volume to mass estimates by measurements of density (Jordan et al. 2006).
The indirect method is employed in this research with the development of density proﬁles in
Chapter 4. Also in Chapter 4, three diﬀerent models are developed for cumulative dominant
stem and whole-tree mass proﬁles. The ﬁrst model is a new cumulative mass proﬁle that is
compared to two common approaches found in the literature: the density-integral (Parresol
1999, Jordan et al. 2006) and the constant density (Chave et al. 2005, Van Deusen and Roesch
2011, Woodall et al. 2011) approaches. The subsequent chapter outlines the methods and
study area of the data used for the development of the dominant stem and whole-tree models.

2

CHAPTER 2
METHODS

2.1 Dominant Stem Volume
The dominant stem in this study was deﬁned by following the largest and most vertical
branch at each fork that was in line with the stem prior to branching. For each tree,
circumference was measured at stump height, 37 cm above ground level, followed by breast
height, 137 cm above ground level. Additional circumference measurements were taken
at 2-m length intervals up to the ﬁrst branch junction, and then continued for all branch
junctions until reaching the terminal bud. Upon encountering the ﬁrst branch, circumference
measurements were taken at the before fork (BF) location of the stem and then at the after
fork (AF) location of each branch based upon functional branch analysis protocols (van
Noordwijk and Mulia 2002). The BF was deﬁned as the location where included bark due
to branching was no longer visible as this is most likely the origin of the branches oﬀ the
dominant stem. The AF was the location immediately after the point at which the fork
occurred. The length from the BF to the AF was also recorded to measure the total path
length from base to tip along the dominant stem. It is recognized that the dominant stem
total path length may be an over-estimate of standing total tree height due to the curvature
of the stem. All circumference measurements were divided by the constant π to convert to
diameter measurements.
The volume of a section f of the dominant stem was determined by Smalian’s formula
for volume:
vf =

A1f + A2f
∗ z,
2

(2.1)

where vf is the volume of section f , in m3 , A1 is the cross-sectional area, in m2 , of the lower
end of section f , A2 is the cross-sectional area, in m2 , of the upper end of section f , and z,

3

in m, is the length of section f (Avery and Burkhart 2002).
The cumulative dominant stem volume proﬁle, vDOM (h), was found by iteratively evaluating
n(h)

vDOM (h) =

vf ,

(2.2)

f =0

where n is the number of segments to a measured height, h, and v0 is the volume of the
stump section. The volume of the stump section was approximated by a cylinder, ignoring
the roots near ground level. The total dominant stem volume was determined by summing
n segments up to total tree height, H. Figure 2.1 shows a graphical representation of the
measurements taken.

2.2 Whole-tree Volume — Random Branch Sampling
An estimate of the total aboveground volume of an entire tree outside bark, VW T , was
obtained by the method of random branch sampling (Gregoire et al. 1995). The same
measurement methods as for the dominant stem were used, but at each branch junction a
segment was selected based on probability proportional to the size of the AF diameter. Based
on these probabilities, the selected section either continues to follow the dominant stem or
diverges following a diﬀerent path through the tree until reaching the tip of a branch. The
probability, pq , of selecting a particular AF was determined by
d2 q
AF

pq = r

(2.3)

d2 q
AF
q=1

where dAF q is the diameter of the qth AF, and r is the number of AF at that particular
branch junction. A random number between 0 and 1 was then used to select the path.
For example, the ﬁrst branch junction shown in Figure 2.1 had three AFs with probabilities of selection of 0.5, 0.3, and 0.2 for AF1 , AF2 , and AF3 , respectively. If the random
number generated was less than or equal to 0.5 then AF1 was selected, between 0.5 and
including 0.8 then AF2 was selected, and greater than 0.8 then AF3 was selected.
4

First Branch Junction:

AF1

CH

L

AF3

BF

AF2
z

Figure 2.1: A simpliﬁed hardwood tree schematic where the dominant stem is indicated by
horizontal lines, the ﬁrst random branch path is indicated by a checkered pattern, the
second random branch path is indicated by dots, and non-sampled branches are solid. The
ﬁrst branch junction at crown height (CH) demonstrates the length (z) and circumference
measurements taken at each branch junction for the before fork (BF) and the after fork
(AF) locations.

5

The sectional volume for the random path was used to compute the volume of the wholetree, using the inverse of the cumulative selection probability as expansion factors for a
section in the random path (Valentine et al. 1984, Gregoire et al. 1995). As seen in Figure
2.1, possible random branch paths could follow any of the ﬁrst order branches oﬀ of the
dominant stem, and then the path continued to be chosen as higher order branches were
encountered until terminating at the tip of a branch. The volume of all branches in the tree
was then calculated by subtracting the total dominant stem volume from the estimate of
whole-tree volume. A second random branch path was sampled on each tree to determine
mean whole-tree volume and the variation of whole-tree volume estimates between the two
diﬀerent selected paths.
To describe the distribution of branch volume in the tree, the mean total branch volume,
VB , determined from the two random paths, was then distributed back to each ﬁrst order
branch, vB l , immediately oﬀ the dominant stem by
d2 l
B
vB l = VB ∗
k
d2 l
B

(2.4)

l=1

where dB l is the diameter of the lth ﬁrst order branch, and k is the total number of ﬁrst order
branches in an individual tree. The volume for each ﬁrst order branch and all attached higher
order branches were incorporated into the proﬁle at the relative height of the associated AF
of the dominant stem (Fig. 2.2). Large amounts of volume can possibly be added at relative
height locations depending on the number of branches and branch sizes at the respective
location. The re-distribution of the branch volume ensured that the ﬁnal observation of the
cumulative whole-tree volume proﬁle was equal to the mean total whole-tree volume outside
bark from the two random branch paths. From these measurements, models for dominant
stem and whole-tree volume were developed in Chapter 3.

6

0

1

RCH

Relative height along dominant stem

Figure 2.2: A simpliﬁed hardwood tree schematic showing the location of branch volume or mass accumulation along the
dominant stem as a function of relative height, ranging from zero at base to one at top of the tree. The ﬁrst branches
encountered are at relative crown height (RCH).

7

2.3 Density Sampling
2.3.1 Tree Density Sampling
For each tree, discs were harvested at every circumference measurement location below crown
height. Above crown height, discs were collected at every AF along the dominant stem
path. The dominant stem path is typically studied for longitudinal density proﬁles. Discs
were only collected at each AF to reduce the likelihood of sampling discs with more than
one pith; however, some sampled discs with an AF in close proximity to the next BF had
multiple piths. Approximately 5- and 10-cm-thick discs were harvested for smaller and larger
circumferences, respectively. Each disc was measured for green mass and green volume
immediately after transportation to the laboratory. Mean disc thickness was calculated by
averaging the thickness of eight locations at 45◦ angles. Green volume was calculated as the
product of mean disc thickness and cross-sectional area of the disc. After being oven-dried
at 105◦ C for 5 d, which was determined to be the length of time until constant mass was
reached (Fig. 2.3), each disc was re-weighed to calculate moisture content, dry mass, and
basic speciﬁc gravity (Williamson and Wiemann 2010). The smaller diameter discs were dried
for 24 h as these discs reached a constant mass in less time than the larger diameter discs
(Fig. 2.3). The basic density, in units of kg/m3 , of each disc was estimated by multiplying
the basic speciﬁc gravity of the disc by the basic density of water, 103 kg/m3 . From these
disc samples, models were developed for the vertical density of the dominant stem (see Ch.
4).
The second random branch path chosen was used to determine if diﬀerences in density
exist between the dominant stem path and any other path that could have been chosen. Tree
disc samples were harvested at every AF for the random branch path to reduce the cost of
sampling. The random branch path discs were processed using the same protocols previously
outlined for the dominant stem discs. From these disc samples, models were developed for
the density of the random branch path (see Ch. 4).

8

30
q

a

q

b

8

25
Mass (kg)

Mass (kg)

6
20
q

q

q

q

q

4

q

15

2

10

0
0

1

2

3

4

5

6

7

0

1

Time (days)
1.2

2

3

4

5

6

7

Time (days)
32

q

q

c

d

1.0
24
q

q

Mass (g)

Mass (kg)

0.8

q

0.6

q

q

16

0.4
8
0.2
0.0

0
0

1

2

3

4

5

6

7

0

Time (days)

4

8

12

16

20

24

Time (hours)

Figure 2.3: The drying time for discs sampled from an American beech tree with a
diameter at breast height, DBH, of 75.4 cm. (a) The oven-drying time for large discs with
diameter outside bark, dob, greater than 50.0 cm. (b) The oven-drying time for medium
discs with dob ranging from 20.1 to 50.0 cm. (c) The oven-drying time for small discs with
dob ranging from 5.1 to 20.0 cm. (d) The oven-drying time for very small discs with dob
ranging from 0.1 to 5.0 cm. Each line and symbol represents a diﬀerent sampled disc in the
respective dob size class.

9

2.3.2 Bark Density Sampling
Tree basic density measurements (above) are a combination of wood and bark. Therefore,
additional measurements of bark were taken to compare bark density to tree density. For
a randomly selected subset of trees that included one tree for each species from each 10-cm
DBH class, bark samples approximately 2 x 2 cm were removed from oven-dried discs with
a chisel and re-hydrated for 24 h in distilled water. The re-hydrated bark samples were
then measured for green volume. For samples that sank during the re-hydration process,
the suspension method was used (Hughes 2005). For bark samples that did not sink during
the re-hydration process, the water displacement method was applied using a needle instead
(Martinez-Cabrera et al. 2009). After green volume was determined, the bark samples were
dried for 24 h at 105◦ C to get oven-dry mass. Basic density for the bark was calculated in
the same manner as tree density from previous discs. Due to the diﬃculty of removing bark
from small discs (diameter outside bark < 4.0 cm), bark samples were not taken along the
entire path length for both dominant stem and random branch paths of the selected subset
of trees.

2.4 Mass Sampling
2.4.1 Dominant Stem Mass Sampling
The dominant stem sectional mass was determined by the product of dominant stem sectional
volume (eq. 2.1) and basic densities from sampled discs. The basic density of a section was
calculated by a weighted average with cross-sectional area as the weight of basic density for
the disc from the top of the section and the previous section’s disc. The mass of the stump
section was approximated by the product of stump sectional volume and density of the DBH
disc. Mass of the top section was approximated by the product of the top sectional volume
and density of the last disc sampled. The cumulative dominant stem mass proﬁle was found
similarly to volume by substituting mass for volume in equation 2.2.
10

2.4.2 Whole-tree Mass — Random Branch Sampling
An estimate of the total aboveground mass of an entire tree outside bark, MW T , was obtained
similarly to whole-tree volume but using only the second random branch path. For each
sectional volume in the random branch path, a weighted average of random branch path
basic densities were used to convert the measurements to mass, analogous to the dominant
stem mass sampling. Whole-tree mass in this study was deﬁned as the mass of the dominant
stem and branches excluding leaves. The total mass of branches was calculated similar
to volume by subtracting the total dominant stem mass from the estimate of whole-tree
mass. Total branch mass was distributed back to each ﬁrst order branch immediately oﬀ
the dominant stem as for volume (eq. 2.4). Figure 2.2 conceptualizes the re-distribution
of branch mass to a proﬁle of whole-tree mass (similar to a whole-tree volume proﬁle) as
a function of relative height. From these measurements, models for dominant stem and
whole-tree mass were developed in Chapter 4.

2.5 Study Site
The study site is 20.8 ha of a 36-ha second-growth maple-beech stand at the Fred Russ
Experimental Forest in southwestern Michigan owned by Michigan State University. The
Fred Russ Experimental Forest is located in Cass County, in Decatur, Michigan (Fig. 2.4),
with a total area of 381 ha that supports a diverse range of species and stand conditions.
Kalamazoo, Ormas, and Oshtemo are the three primary soil series found at the site. The
Kalamazoo and Oshtemo soil series are ﬁne-loamy and coarse-loamy, mixed, mesic typic hapludalfs, respectively. The Ormas soil series is a coarse-loamy, mixed, mesic arenic hapludalfs
on an outwash plain landform with a level topography (Soil Survey Staﬀ 2011). Average
annual precipitation over 30 years is 1035 mm (NCDC 2011).
In the spring of 2010, a windstorm uprooted more than 200 trees in the stand. After
2

salvaging the wind-thrown trees, the residual basal area of the stand was 22.13 m . No
ha

11

inventory data for this stand was available prior to the windstorm, so caution should be
taken when generalizing from the models developed from wind-thrown trees to the residual
stand. Tables 2.1 and 2.2 summarize the average number of trees per ha and mean basal
2

area in m by species and DBH class for the residual stand, respectively.
ha

12

A

^

/

!
!
!
!
!!
!
!
!
!
!
!! !

1 cm equals 183 m

!
!

Legend

^
!

!
!
!
!!
!
!

B

Forest Entrance
Trees
Plots

/

Study Area

Forest Boundary
Forest Stands

^

1 cm equals 112.62 km

Figure 2.4: (A) Shows the location of the 20.8-hectare research area in the 36-hectare
second-growth maple-beech stand, ﬁxed-area plots, and sampled trees within the Fred Russ
Experimental Forest. (B) The star represents the location of the entrance to the Fred Russ
Experimental Forest in Cass County, Michigan.

13

Table 2.1: Summary of the mean number of trees per ha by species and 10 cm diameter at breast height (DBH) class for the
residual stand.
Species
American beech (Fagus grandifolia Ehrh.)
Sugar maple (Acer saccharum Marsh.)
Black cherry (Prunus serotina Ehrh.)
American basswood (Tilia americana L.)
Slippery elm (Ulmus rubra Muhl.)
American hophornbeam (Ostrya virginiana Mill.)
Tulip-poplar (Liriodendron tulipofera L.)
White ash (Fraxinus americana L.)
All

10
8.55
21.38
—
—
1.43
1.43
—
—
32.79

20
4.28
35.17
0.48
—
0.95
—
0.48
—
41.26

14

30
4.75
28.04
—
0.48
—
—
—
0.48
33.75

DBH Class (cm)
40
50
60
3.33
5.70
5.70
22.81 15.21 3.80
—
—
—
0.48
—
—
—
—
0.48
—
—
—
—
—
0.48
—
—
—
26.62 20.91 10.46

70
4.28
1.43
—
—
0.48
—
—
—
6.19

80
3.33
1.90
—
0.48
—
—
0.48
—
6.19

90
2.38
0.48
—
—
—
—
0.48
—
3.34

Total
42.30
130.21
0.48
1.44
3.34
1.43
1.92
0.48
181.60

2

Table 2.2: Summary of the average basal area in m by species and 10 cm diameter at breast height (DBH) class for the
ha
residual stand.
Species
American beech (Fagus grandifolia Ehrh.)
Sugar maple (Acer saccharum Marsh.)
Black cherry (Prunus serotina Ehrh.)
American basswood (Tilia americana L.)
Slippery elm (Ulmus rubra Muhl.)
American hophornbeam (Ostrya virginiana Mill.)
Tulip-poplar (Liriodendron tulipofera L.)
White ash (Fraxinus americana L.)
All

10
0.09
0.27
—
—
0.02
0.02
—
—
0.40

20
0.12
1.13
0.01
—
0.04
—
0.02
—
1.32

15

30
0.37
1.96
—
0.04
—
—
—
0.03
2.40

DBH Class (cm)
40
50
60
70
0.43 1.15 1.57 1.59
2.92 2.98 1.04 0.59
—
—
—
—
0.07 —
—
—
—
— 0.14 0.17
—
—
—
—
—
— 0.15 —
—
—
—
—
3.42 4.13 2.90 2.35

80
1.62
0.94
—
0.25
—
—
0.25
—
3.06

90
1.55
0.33
—
—
—
—
0.29
—
2.17

Total
8.47
12.16
0.01
0.36
0.37
0.02
0.71
0.03
22.13

2.6 Tree Selection
A sample of 32 wind-thrown trees were measured with intact crowns from the population
of uprooted trees. Sampling began in late spring of 2010 and included as wide a range of
DBH as possible, irrespective of species, with at least one individual in each 10 cm DBH
class ranging from 10 - 90 cm classes (Fig. 2.5). Figure 2.5 shows the relative frequency of
the DBH distribution for the residual stand in 10 cm classes. Selected trees were relatively
isolated from other wind-thrown trees so that the branches were easily measurable. The
sample trees had a mean DBH of 50.9 cm and a mean height of 29.4 m with 87.5 percent
of the trees being either American beech (Fagus grandifolia Ehrh.) or sugar maple (Acer
saccharum Marsh., Table 2.3). Table 2.3 lists the summary statistics by species of sample
size, DBH, and height of the 32 sample trees.

16

0.025
0.020
0.015
0.010
0.000

0.005

Density

20

40

60

80

DBH (cm)

Figure 2.5: Histogram of the relative frequency of the residual stand, represented by
diagonal line ﬁlled bars, and the sampled trees, represented by gray ﬁlled bars, divided into
10 cm diameter at breast height (DBH) classes.

17

Table 2.3: Species, code, sample size (n), diameter at breast height (DBH), and height (H) summary statistics for the data set.
Species
Code
American beech (Fagus grandifolia Ehrh.) FG
Sugar maple (Acer saccharum Marsh.)
AS
Pignut hickory (Carya glabra Mill.)
CG
Black cherry (Prunus serotina Ehrh.)
PS
American basswood (Tilia americana L.) TA
Slippery elm (Ulmus rubra Muhl.)
UR
All
ALL

n
15
13
1
1
1
1
32

Mean DBH (cm) [SD, min, max] Mean H (m) [SD, min, max]
56.6 [19.5, 15.3, 91.1]
30.8 [4.9, 16.5, 36.1]
44.6 [12.4, 27.5, 63.2]
28.4[3.3, 22.2, 34.0]
62.6 [NA, NA, NA]
34.5 [NA, NA, NA]
23.0 [NA, NA, NA]
19.1 [NA, NA, NA]
41.0 [NA, NA, NA]
24.0 [NA, NA, NA]
71.5 [NA, NA, NA]
33.3 [NA, NA, NA]
50.9 [17.6, 15.3, 91.1]
29.4 [4.7, 16.5, 36.1]

18

CHAPTER 3
CUMULATIVE VOLUME PROFILES FOR DOMINANT STEMS AND
WHOLE TREES

3.1 Related Dominant Stem Taper and Volume Modeling
3.1.1 Dominant Stem Taper Modeling
Taper modeling is generally deﬁned as the ﬁtting of a mathematical function that graphically
represents the form of the observed dominant stem. Foresters widely use taper models
because diameter at any point along the dominant stem of a tree can be predicted from a
few predictor variables measured on a tree. The most commonly used predictor variables
are DBH and total height of a tree. As merchantability standards have changed over time,
taper models have been ﬂexible enough to account for the changes in minimum commercially
accepted upper stem diameters. Numerous taper models have been developed to describe
the dominant stem of speciﬁc species, and only a selection of these studies are reviewed here.
The taper model examples that follow were chosen based their relevance to this current
research.
Many taper models have been formulated for the dominant stem of a tree, as this portion
is the simplest to model and has the greatest economic value. Conceptually, the dominant
stem is divided into three distinct sections: the lower, the middle, and the upper section.
Typically, the geometric shape of the lower section is modeled as a neiloidal shape, the middle
section follows a paraboloidal shape, and the top section follows a conical shape (Newnham
1992). Max and Burkhart (1976) applied segmented polynomial functions to each of these
diﬀerent sections and joined them together with indicator variables. On the other hand,
Kozak (1988) used a variable-exponent taper model in which a single continuous function
was used and only the exponent varied depending on the location along the stem.

19

In the previous examples and in taper models in general, the eﬀect of branching on the
dominant stem has been largely ignored (but see Adu-Bredu et al. 2008). However, it is
recognized that crown variables (e.g., crown height, crown length, or crown ratio) eﬀect
taper (Larson 1963, Valentine and Gregoire 2001, Leites and Robinson 2004). For example,
Valentine and Gregoire (2001) use a combined variable-exponent and variable-form taper
model with the height to the base of the live crown as a predictor. Leites and Robinson
(2004) found crown length and crown ratio of an individual tree to improve the ﬁt of the
dominant stem taper models being tested.
One study directly dealt with the eﬀects of branching on dominant stem taper models;
Adu-Bredu et al. (2008) studied the stem proﬁle of teak (Tectona grandis, L.f.) in West
Africa. Their model explicitly accounted for up to two major forks branching oﬀ the dominant stem. Their proﬁles showed distinct reduction in the dominant stem proﬁle whenever
branching occurs. This model accounted for the eﬀect of branching on the dominant stem
but did not account for the volume of the branches.
Mixed-eﬀects taper modeling is prevalent in the literature because of the hierarchical
structure of the data with measurements taken within an individual tree and between trees
(Valentine and Gregoire 2001, Leites and Robinson 2004, Westfall and Scott 2010). Valentine and Gregoire (2001) employed mixed-eﬀects modeling to incorporate the correlation of
multiple measurements on the same tree, and found the inclusion of random eﬀects gave
better ﬁts to individual trees compared to ﬁxed eﬀects only. As measurements taken closer
together within a single tree are more likely to be similar that distant measurements on the
same tree. Westfall and Scott (2010) modiﬁed the model developed by Valentine and Gregoire (2001) and applied it to commercial tree species of the northeastern U.S. The modiﬁed
model performed better than the original by re-arranging the random eﬀects parameters and
estimating the join points rather than explicitly using the height of the base of the live crown.
Leites and Robinson (2004) applied random eﬀects to an existing taper model to improve
model ﬁt and determined additional predictor variables explained additional variation that

20

was not originally considered in the taper model. Due to the hierarchical nature of the data,
this research also employs a mixed-eﬀects modeling framework.
A ﬁnal important principle of continuous taper models is that the volume of the dominant
stem can be predicted, or even segments of the stem to a speciﬁc limiting diameter or height
via integration (Jordan et al. 2006). Conversely, the ﬁrst derivative of a cumulative volume
model is equivalent to a taper model. The current research uses the later principle to develop
dominant stem and whole-tree cumulative volume models. The cumulative volume models
are then applied to the taper modeling paradigm for the dominant stem and whole-tree so
that dominant stem and whole-tree taper can be calculated with respect to relative height.

3.1.2 Volume Modeling
Total volumes have long been estimated by standard mensuration techniques for the dominant stem of trees (Avery and Burkhart 2002). The DBH and total tree height are the
two most common predictor variables for total dominant stem volume (Hahn and Hansen
1991). Many diﬀerent volume models have been developed across diﬀerent regions of the
U.S. (Woodall et al. 2011).
Researchers have not only been interested in total volume, but cumulative volume proﬁles at diﬀering heights (Gregoire and Schabenberger 1996a;b, Fonweban et al. 2012). In
these works, only the dominant stem cumulative volume proﬁle was examined. The current research expands from the dominant stem only to the inclusion of a whole-tree volume
proﬁle.
Similar to taper modeling, mixed-eﬀects modeling is prevalent for cumulative volume
proﬁles (Gregoire and Schabenberger 1996a, Fonweban et al. 2012). Gregoire and Schabenberger (1996a) developed a cumulative dominant stem volume model using non-linear mixed
eﬀects to account for the spatial correlation of multiple measurements on an individual tree.
They found that the mixed-eﬀects model ﬁt the individual trees better than the ﬁxed-eﬀects
model, which did not account for within-tree correlation. However, the model ﬁt was poor

21

for both mixed and ﬁxed eﬀects for the region of the stem below DBH. The follow-up to
this study, by Gregoire and Schabenberger (1996b), re-deﬁned parameters in the cumulative volume model to achieve a better ﬁt at lower heights of the dominant stem. Fonweban
et al. (2012) also found the mixed-eﬀects modeling approach gave an improved ﬁt over the
ﬁxed-eﬀects models. They also modeled the covariance structure of the mixed-eﬀects model
and determined an increase in the residual variance of the models was present, so they recommended a simple unstructured covariance structure of the random eﬀects would suﬃce.
Due to the hierarchical nature of the data, this research employs a mixed-eﬀects modeling
framework with the assumption of an unstructured covariance structure.
Research on the relative volume of branches has received much less attention than the
dominant stem (MacFarlane 2011). The allocation of aboveground volume relationships
between tree volume and branch volume have recently been compared among similar trees
with and without signiﬁcant branching by MacFarlane (2010). In this work, the displacement
of dominant stem volume to branches has been expanded by including the largest branches
in the development of a varying-centroid method to predict volume ratios between the wholetree and the dominant stem. The best predictor variables of these ratios were the interaction
between crown ratio and diameter of largest branch, and then the interaction between crown
ratio and DBH. These results indicate that measurements of crown variables are needed to
link the estimation of whole-tree volume to dominant stem volume.
Allometric scaling principles have also recently been used to relate large branch volume to
dominant stem volume to estimate whole-tree volume. MacFarlane (2011) collected volume
data of 11 common hardwood species in Michigan. Volume expansion factors, deﬁned as
the ratio of branch-plus-bole to bole volume, were used to estimate the whole-tree volume.
It was found that, beyond the usual metrics of DBH and height for inventorying a tree,
crown variables were also necessary parameters for accurate calculation of volume expansion
factors. Once again, the best predictor variables were found to be the diameter of the largest
branch and the crown ratio, indicating that crown variables are important for estimation of

22

whole-tree volume. However, a minimum upper branch diameter of 9 cm was imposed in
the collection of the branch volume data. This research expands upon this previous study
by not limiting branch diameter in estimating whole-tree volume.
The primary objective of this chapter is to develop a new vertical proﬁle model to relate whole-tree volume to dominant stem volume, which can estimate component-wise tree
attributes of dominant stem and branches. The cumulative volume proﬁles developed here
build upon previous research by expanding dominant stem models to the whole aboveground
tree. The ﬁrst derivative of the cumulative volume proﬁles is the taper model, and it will be
combined with density proﬁles for mass modeling in Chapter 4.

3.2 Models
The data used for model development were trees sampled from a second-growth maple-beech
stand at Fred Russ Experimental Forest (see Ch. 2). A ﬁxed-eﬀects model (shown below,
see eqs. 3.1 and 3.2) was selected for each individual tree with parameters estimated using
the nlsList function from the nlme package (Pinheiro et al. 2011). The parameters from
the ﬁxed-eﬀects model were used as starting values for the mixed-eﬀects models using the
nlme function in the nlme package. The models were ﬁt to all species, American beech, and
sugar maple with the best-ﬁt model being selected based on the lowest Aikaike’s Information
Criterion, AIC, value. American beech and sugar maple were ﬁtted to the model individually
as multiple observations of each of these species were present. Models were also selected based
on having the smallest mean relative diﬀerence in relation to estimated total volumes. All
statistical models were developed in the R statistical environment (R Development Core
Team 2010).

3.2.1 Dominant Stem Cumulative Volume Models
Non-linear regression modeling was used in developing the cumulative volume proﬁles for
dominant stems in terms of relative height. The base ﬁxed-eﬀects model is modiﬁed from
23

the relative dominant stem volume model of MacFarlane (2010):
vDOM = VDOM ∗ (1 − (1 − rh)α1 ) + ,

(3.1)

where vDOM is the dominant stem cumulative volume in m3 at a relative height (rh), VDOM
is total dominant stem volume in m3 , α1 is a parameter to be estimated, and

is an error

term.
3.2.2 Whole-tree Cumulative Volume Models
The whole-tree cumulative volume model uses the modiﬁed relative dominant stem volume
model (eq. 3.1) for the portion below relative crown height with the addition of another
model, where the intercept is shifted to relative crown height for the portion above relative
crown height:
vAC = VRCH + βV ∗ (1 − (1 − (rh − RCH))α2 ) + ,

(3.2)

where vAC is the dominant stem plus branches cumulative volume in m3 at a relative height
(rh), VRCH is total tree volume at relative crown height (RCH) in m3 , βV is a either a ﬁxed
parameter or an estimated parameter, α2 is a parameter to be estimated, and

is an error

term. In the case where βV is ﬁxed, the new model has the same form as model 3.2 setting
V
− VRCH
,
βV = W T
1 − RCH α2

(3.3)

where VW T is the total whole-tree volume in m3 .
Models 3.1 and 3.2 or models 3.1 and 3.3 are combined to get the whole-tree cumulative
volume proﬁle as
vW T = I1 vDOM + I2 vAC ,

(3.4)

where vW T is the whole-tree cumulative volume in m3 at a relative height, I1 is an indicator
variable with a value of 1 when relative height is below relative crown height and a value of
0 when relative height is above relative crown height, and I2 is an indicator variable with
values of 0 when relative height is below relative crown height and 1 when relative height is
above relative crown height.
24

3.2.3 Centroid of Dominant Stem and Whole-tree Volume Models
The centroid of volume is deﬁned as the location aboveground in a tree where half of the
total volume is above and half of the total volume is below the location (Forslund 1982). This
location along the dominant stem has been shown to be an accurate and precise estimate of
dominant stem volume requiring a single diameter measurement (Wood et al. 1990, Wiant
et al. 1991). MacFarlane (2010) expands the theory of the centroid method to the whole tree.
The centroid of dominant stem volume is found by re-arranging the cumulative dominant
stem volume (model 3.1) in terms of relative volume, setting relative volume equal to a half,
and solving for relative height, similar to equation 4 in MacFarlane (2010):
1

rhDOM.c = 1 − 0.5 α1 ,

(3.5)

where rhDOM.c is the centroid of dominant stem volume, and α1 is the estimated parameter
from model 3.1.
The centroid of whole-tree volume has two possible solutions due to the segmented model
(eq. 3.4). The ﬁrst solution occurs when the centroid of whole-tree volume occurs in the
ﬁrst segment of the model, below relative crown height:
rhW T.c = 1 − (1 −

1
VW T α
) 1,
2VDOM

(3.6)

where rhW T.c is the centroid of whole-tree volume, VW T is the total whole-tree volume,
VDOM is the total dominant stem volume, and α1 is the estimated parameter from model
3.1. Otherwise, the second solution occurs when the centroid of whole-tree volume occurs in
the second segment of the model, above relative crown height:
1
V
V
rhW T.c = 1 + RCH − (1 − W T + RCH ) α2 ,
2βV
βV

(3.7)

where RCH is relative crown height, βV is the ﬁxed parameter from model 3.3, VRCH is
the volume at relative crown height, and α2 is the estimated parameter from model 3.2.

25

3.3 Results
3.3.1 Total Volume Estimates
Table 3.1 shows the summary statistics for the total volume of dominant stems and whole
trees by species and for all trees in the dataset. The total dominant stem volume and mean
total whole-tree volume estimates are shown in Table 3.2 for each individual. Table 3.2
also shows the two whole-tree volume estimates for each tree by random branch path, the
associated standard deviation, and coeﬃcient of variation around the mean whole-tree total
volume. The smallest coeﬃcient of variation of 0.01 was found in the 40 cm DBH class,
and the maximum coeﬃcient of variation of 0.49 was found in the largest DBH, 90 cm,
class. The average coeﬃcient of variation of whole-tree volume for all trees was 0.11 with
20 of the 32 trees having a coeﬃcient of variation of less than or equal to 0.10. However,
a larger coeﬃcient of variation was present in both the small and large DBH classes. The
total volume of both dominant stems and whole trees showed an increasing pattern for larger
DBH for all species, American beech, and sugar maple (Fig. 3.1).
Table 3.1: Species code (spp), number of trees (n), mean dominant stem volume (VDOM ),
mean total volume (VW T ) with standard deviation (SD), minimum (min), and maximum
(max) values by species.
spp
FG
AS
CG
PS
TA
UR
ALL

n
15
13
1
1
1
1
32

Mean VDOM (m3 ) [SD, min, max] Mean VW T (m3 ) [SD, min, max]
3.70 [2.32, 0.14, 8.06]
8.04 [7.87, 0.23, 33.17]
2.27 [1.36, 0.74, 4.86]
3.85 [2.53, 1.06, 8.84]
5.16 [NA, NA, NA]
8.60 [NA, NA, NA]
0.36 [NA, NA, NA]
0.50 [NA, NA, NA]
1.47 [NA, NA, NA]
2.00 [NA, NA, NA]
6.39 [NA, NA, NA]
13.85 [NA, NA, NA]
3.07 [2.10, 0.14, 8.06]
6.11 [6.17, 0.23, 33.17]

26

Table 3.2: Species code (spp), diameter at breast height (DBH), DBH class (Dclass), total
dominant stem volume (VDOM ), total volume estimate from the ﬁrst random branch
sample (VW T 1 ), total volume estimate from the second random branch sample (VW T 2 ),
mean total volume (VW T ), standard deviation of total volume (SD), and coeﬃcient of
variation of total volume (CV) summary statistics for each individual tree.
spp
FG
PS
AS
AS
AS
AS
AS
FG
AS
TA
AS
FG
FG
FG
FG
AS
AS
FG
AS
FG
FG
AS
AS
CG
AS
FG
FG
UR
FG
FG
FG
FG

DBH Dclass
15.3 10
23.0 20
27.5 30
31.7 30
33.1 30
33.9 30
35.6 40
36.7 40
38.8 40
41.0 40
42.8 40
42.9 40
45.1 40
46.0 50
47.1 50
48.1 50
48.5 50
50.1 50
54.6 50
55.1 50
56.3 60
60.4 60
61.9 60
62.6 60
63.2 60
63.6 60
67.5 70
71.5 70
75.4 80
78.2 80
78.5 80
91.1 90

VDOM
0.14
0.36
0.74
0.82
1.13
0.95
1.29
1.25
1.54
1.47
2.39
1.86
2.73
2.51
2.42
2.26
2.79
2.10
2.85
3.32
3.77
3.49
4.86
5.16
4.35
4.34
3.91
6.39
4.47
7.04
7.61
8.06

VW T 1
0.24
0.51
1.11
1.71
1.56
1.73
2.03
1.98
2.19
2.01
2.89
3.25
3.71
3.46
3.95
4.41
4.64
4.68
5.32
4.87
6.81
7.05
7.53
8.77
8.47
6.63
6.46
13.66
8.86
12.31
11.45
21.75

VW T 2
0.23
0.49
1.01
1.21
1.91
2.21
2.24
2.64
2.05
1.99
2.92
3.52
5.40
4.38
4.46
3.82
4.12
6.13
4.66
5.87
7.04
6.82
7.28
8.44
9.21
7.42
8.63
14.04
11.74
16.83
11.83
44.58

27

VW T
0.23
0.50
1.06
1.46
1.73
1.97
2.13
2.31
2.12
2.00
2.90
3.38
4.55
3.92
4.21
4.11
4.38
5.41
4.99
5.37
6.92
6.94
7.40
8.60
8.84
7.03
7.55
13.85
10.30
14.57
11.64
33.17

SD of VW T
0.01
0.01
0.07
0.35
0.25
0.34
0.14
0.47
0.10
0.01
0.02
0.19
1.20
0.65
0.36
0.42
0.37
1.03
0.47
0.71
0.16
0.17
0.17
0.24
0.52
0.56
1.54
0.27
2.04
3.20
0.27
16.14

CV of VW T
0.03
0.03
0.07
0.24
0.14
0.17
0.07
0.20
0.05
0.01
0.01
0.06
0.26
0.17
0.09
0.10
0.08
0.19
0.09
0.13
0.02
0.02
0.02
0.03
0.06
0.08
0.20
0.02
0.20
0.22
0.02
0.49

35
WT
DOM

30
25
20
15
10
5

ALL

0

Total Volume ( m3)

0

20

40

60

80

100

35
30
25
20
15
10
5

FG

0
0

20

40

60

80

100

8
6
4
2
AS

0
0

20

40

60

80

100

DBH (cm)
Figure 3.1: The total volume, in m3 , of sampled trees for the dominant stem (DOM) and
whole tree (WT) versus diameter at breast height (DBH), in cm, for all trees (ALL),
American beech (FG), and sugar maple (AS).

28

3.3.2 Cumulative Volume Proﬁles
The models, below, added random eﬀects to models 3.1 and 3.2 to create mixed-eﬀects
models. The model selected for the cumulative volume of the dominant stem applied to all
species, American beech, and sugar maple is
α +u
vDOMij = VDOMj ∗ (1 − (1 − rhij ) 1 1j ) + ij ,

(3.8)

where vDOMij is the cumulative volume of stem position i for tree j in m3 , VDOMj is total
dominant stem volume in m3 , rhij is relative height of position i for tree j, α1 is a ﬁxed
eﬀect parameter to be estimated, u1j is a random eﬀect parameter to be estimated, and ij is
a random error term. Table 3.3 gives the parameter estimates of model 3.8. The all-species
model was selected as the best model because it had the lowest AIC of the three models, and
the total dominant stem model was compatible with the ﬁnal observation of total dominant
stem volume from Smalian’s formula, as the mean relative diﬀerence is equal to zero (to
three decimal places).
Table 3.3: Parameter estimates of model 3.8 for the cumulative volume of the dominant
stem for the all-species (ALL), American beech (FG), and sugar maple (AS) mixed-eﬀects
models. For the ﬁxed parameters, the standard error of estimates is given in parantheses.
Parameter
α1
var(u1j )
ij

AIC
Mean Relative Diﬀerence

ALL Estimate FG Estimate
2.783 (0.048)
2.833 (0.068)
0.067
0.062
0.045
0.049
-6479.677
-3199.169
0.000
0.000

AS Estimate
2.782 (0.076)
0.072
0.030
-2955.086
0.000

Two potential mixed-eﬀects models for the portion of the whole-tree model above relative
crown height are given below:
α +u
vACij = VRCHj + (βV + u3j ) ∗ (1 − (1 − (rhij − RCHj )) 2 2j ) + ij ,

(3.9)

where vACij is the whole-tree cumulative volume of stem position i in tree j in m3 , VRCHj
is total tree volume at relative crown height (RCH) in m3 , βV and α2 are ﬁxed eﬀect
29

parameters to be estimated, u2j and u3j are random eﬀect parameters to be estimated, and
ij

is a random error term. In the case where βV is ﬁxed for tree j, the new model has the

same form as model 3.9, except only one random eﬀect parameter (u2j ), with
βV j =

VW Tj − VRCHj
α +u
1 − RCHj 2 2j

,

(3.10)

where VW Tj is the total whole-tree volume for tree j in m3 .
The ﬁnal whole-tree cumulative volume proﬁle (similar to eq. 3.4) now uses models 3.8
and 3.9 or models 3.8 and 3.10 for the segmented model:
vW Tij = I1 vDOMij + I2 vACij ,

(3.11)

where vW Tij is the whole-tree cumulative volume of stem position i in tree j in m3 , I1 is an
indicator variable with a value of 1 when relative height is below relative crown height and a
value of 0 when relative height is above relative crown height, and I2 is an indicator variable
with values of 0 when relative height is below relative crown height and 1 when relative
height is above relative crown height. Tables 3.4 and 3.5 give the parameter estimates for
models 3.9 and 3.10.
Model 3.10 was chosen as the best-ﬁt ﬁnal model of the two for the portion of the
segmented whole-tree cumulative volume model above relative crown height as it had a
lower AIC of -644.505 for the all-species model compared to an AIC of -375.868 for the
all-species model 3.9. The total whole-tree volume for model 3.10 is also compatible with
the total whole-tree volume estimate given by random branch sampling. The all-species
model 3.10 was selected over the American beech model based on AIC; however, the sugar
maple model would be selected over the all-species model using AIC. The all-species model
was selected as the ﬁnal model for whole-tree volume of all trees rather than applying a
separate sugar maple model to maple trees to be compatible with the ﬁrst segment of the
segmented model, as the ﬁrst segment of the model below relative crown height was ﬁt using
an all-species model.

30

Table 3.4: Parameter estimates of model 3.9 for the cumulative whole-tree volume above
relative crown height for the all-species (ALL), American beech (FG), and sugar maple
(AS) mixed-eﬀects models. For the ﬁxed parameters, the standard error of estimates is
given in parantheses.
Parameter
α2
β
var(u2j )
var(u3j )
cov(u2j , u3j )
ij

AIC
Mean Relative Diﬀerence

ALL Estimate
4.328 (0.289)
4.054 (0.879)
2.233
24.683
0.512
0.191
-375.868
0.499

FG Estimate
4.345 (0.407)
5.596 (1.670)
2.177
41.736
0.448
0.242
220.736
0.496

AS Estimate
4.339 (0.418)
2.326 (0.498)
2.104
3.216
-0.156
0.091
-1059.232
0.664

Table 3.5: Parameter estimates of model 3.10 for the cumulative whole-tree volume above
relative crown height for the all-species (ALL), American beech (FG), and sugar maple
(AS) mixed-eﬀects models. For the ﬁxed parameters, the standard error of estimates is
given in parantheses.
Parameter
α2
var(u2j )
ij

AIC
Mean Relative Diﬀerence

ALL Estimate
4.461 (0.274)
2.077
0.195
-644.505
0.000

FG Estimate
4.487 (0.405)
2.184
0.245
80.574
0.000

AS Estimate
4.445 (0.383)
1.789
0.096
-1115.357
0.000

Figures 3.2 and 3.3 show the dominant stem and whole-tree cumulative volume proﬁle
for the (a) minimum, (b) median, and (c) maximum DBH trees of American beech and sugar
maple, respectively. The all-species parameter estimates were used for both the dominant
stem (eq. 3.8) and whole-tree (eq. 3.11) proﬁles for both American beech and sugar maple.
For American beech, the total whole-tree volume increases by a factor of ten in each successive
proﬁle, and the relative crown height also decreases with increasing DBH. A noticeable
increase in branch volume, which is the region between the whole-tree and dominant stem
cumulative volume proﬁles, is also present in each successive proﬁle. The sugar maple proﬁles
show similar patterns to the American beech with total whole-tree volume increasing by
a factor of three in each successive proﬁle. However, the relative crown height appears to

31

increase before decreasing with increasing DBH, and the branch volume also decreases before
increasing with increasing DBH.

a

0.25
0.20
0.15
0.10
0.05
0.00

Cumulative
Volume ( m3)

0.0

3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

b

0.0

30
25
20
15
10
5
0

c

0.0

Relative height along dominant stem
Figure 3.2: Cumulative volume, m3 , proﬁles for the dominant stem (model 3.8),
represented by a dashed line above relative crown height, and whole tree (model 3.11),
represented as the solid line, as a function of relative height along the dominant stem for
the (a) minimum, (b) median, and (c) maximum diameter at breast height (DBH)
American beech (FG) trees. The vertical dashed line represents relative crown height.

32

a

1.0
0.8
0.6
0.4
0.2
0.0

Cumulative
Volume ( m3)

0.0

3.0
2.5
2.0
1.5
1.0
0.5
0.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

b

0.0

c

8
6
4
2
0
0.0

Relative height along dominant stem
Figure 3.3: Cumulative volume, m3 , proﬁles for the dominant stem (model 3.8),
represented by a dashed line above relative crown height, and whole tree (model 3.11),
represented as the solid line, as a function of relative height along the dominant stem for
the (a) minimum, (b) median, and (c) maximum diameter at breast height (DBH) sugar
maple (AS) trees. The vertical dashed line represents relative crown height.

33

3.3.3 Centroid of Dominant Stem and Whole-tree Volume Proﬁles
The centroid of dominant stem volume proﬁle for tree j is similar to model 3.5 with the
addition of a random eﬀect, u1j :
1

α +u
rhDOM.cj = 1 − 0.5 1 1j .

(3.12)

The mean centroid of dominant stem volume for all species was 0.222 (Table 3.6). Across
species, the mean centroid of dominant stem volume ranged between 0.218 for American
beech to 0.251 for pignut hickory. No deﬁnitive pattern for the centroid of dominant stem
volume was seen across 10 cm DBH classes. Crown ratio (CR = 1 - RCH) was also examined
for a potential pattern in the centroid of dominant stem volume (Table 3.7). Crown ratio
classes (CRC) were divided to the nearest tenth with classes ranging from 0.5 to 0.9 indicating
all of the trees had fairly long crowns. Unlike the DBH classes, the centroid of dominant
stem volume showed a decreasing pattern across crown ratio classes.

34

Table 3.6: The mean centroid of dominant stem volume by species and 10 cm diameter at breast height (DBH) classes.
Standard deviation is given in parentheses.
Species
FG

DBH Class (cm)
10
20
0.220
(NA)

AS

30

0.223
( 0.017)

CG
PS

40
50
60
70
0.227
0.222
0.220
0.206
(0.014) (0.017) (0.006) (NA)
0.235
0.214
0.217
(0.016) (0.014) (0.027)
0.251
(NA)

0.222
(NA)

TA

0.241
(NA)

UR
ALL

0.220
(NA)

0.222 0.223
(NA) (0.017)

0.232
(0.014)

35

80
90
All
0.215
0.197 0.218
(0.028) (NA) (0.016)
0.222
(0.018)
0.251
(NA)
0.222
(NA)
0.241
(NA)
0.219
0.219
(NA)
(NA)
0.218
0.223
0.213
0.215
0.197 0.222
(0.015) (0.022) (0.009) (0.028) (NA) (0.017)

Table 3.7: The mean centroid of dominant stem volume by species and crown ratio class
(CRC). Standard deviation is given in parentheses.
CRC
Species 0.5
0.6
0.7
0.8
FG
0.229
0.236
0.208
0.213
(0.011) (0.010) (0.016) (0.006)
AS
0.240
0.225
0.208
0.232
(0.013) (0.012) (0.018) (0.018)
CG
0.251
(NA)
PS
0.222
(NA)
TA
0.241
(NA)
UR
0.219
(NA)
ALL
0.231
0.233
0.208
0.221
(0.012) (0.013) (0.016) (0.013)

0.9
0.201
(NA)

All
0.218
(0.016)
0.222
(0.018)
0.251
(NA)
0.222
(NA)
0.241
(NA)
0.219
(NA)
0.201 0.222
(NA) (0.017)

The centroid of whole-tree volume for tree j has two possible solutions due to the segmented model (eq. 3.11). Similar to the centroid of dominant stem volume proﬁle, the ﬁnal
two solutions for the centroid of whole-tree volume have random eﬀects added, u1j and u2j ,
to models 3.6 and 3.7, respectively:
rhW T.cj = 1 − (1 −

VW Tj
2VDOMj

rhW T.cj = 1 + RCHj − (1 −

VW Tj
2β

1
α1 +u1j
)
,

+

VRCHj
β

1
α2 +u2j
)
.

(3.13)

(3.14)

Twenty-four trees had a centroid of whole-tree volume occur above relative crown height
(eq. 3.14) with the remainder of the trees having centroids below relative crown height
(eq. 3.13). The mean whole-tree centroid of volume for all species is 0.390 (Table 3.8).
Across species, centroids of whole-tree volume ranged from 0.302 for black cherry to 0.499
for slippery elm, which was considerably larger than the centroid of dominant stem. No
distinct pattern was observed for the centroid of whole-tree volume along DBH classes;

36

however, centroids of whole-tree volume clearly decreased with increasing crown ratio class
(Table 3.9).
Figure 3.4 shows the relative cumulative volume proﬁles for the dominant stem and whole
tree for the (a) minimum, (b) median, and (c) maximum DBH American beech trees. In
terms of relative height, the dominant stem relative volume proﬁle will always be greater
than the whole-tree relative volume proﬁle because the denominator of relative volume of
total whole-tree volume will always be larger than dominant stem volume. For the three
beech trees, the mean centroid shift from dominant stem to whole tree was 0.173. This was
similar to the mean centroid shift from dominant stem to whole tree for all trees of 0.178.

37

Table 3.8: The mean centroid of whole-tree volume classiﬁed by species and 10 cm diameter at breast height (DBH) classes
with the standard deviation given in parentheses.
Species
FG

DBH Class (cm)
10
20
0.382
(NA)

AS

30

0.383
( 0.032)

CG
PS

40
50
60
70
0.408
0.403
0.396
0.316
(0.042) (0.101) (0.108) (NA)
0.362
0.393
0.380
(0.051) (0.037) (0.015)
0.419
(NA)

0.302
(NA)

TA

0.364
(NA)

UR
ALL

0.382
(NA)

0.302 0.383
(NA) (0.032)

0.382
(0.045)

38

80
90
All
0.414
0.375 0.396
(0.058) (NA) (0.066)
0.380
(0.033)
0.419
(NA)
0.302
(NA)
0.364
(NA)
0.499
0.499
(NA)
(NA)
0.399
0.392
0.407
0.414
0.375 0.390
(0.075) (0.052) (0.130) (0.058) (NA) (0.056)

Table 3.9: The mean centroid of whole-tree volume by species and crown ratio class
(CRC). Standard deviation is given in parentheses.
CRC
Species 0.5
0.6
0.7
0.8
FG
0.460
0.468
0.378
0.322
(0.052) (0.012) (0.012) (0.008)
AS
0.373
0.410
0.362
0.372
(0.073) (0.022) (0.011) (0.016)
CG
0.419
(NA)
PS
0.302
(NA)
TA
0.364
(NA)
UR
0.499
(NA)
ALL
0.438
0.425
0.370
0.335
(0.070) (0.039) (0.014) (0.031)

39

0.9
0.303
(NA)

All
0.396
(0.066)
0.380
(0.033)
0.419
(NA)
0.302
(NA)
0.364
(NA)
0.499
(NA)
0.303 0.390
(NA) (0.056)

1.0

a

0.8
0.6

DOM.c
RCH
WH.c

0.4
0.2
0.0

Relative Volume

0.0

1.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

b

0.8
0.6
0.4
0.2
0.0
0.0

1.0

c

0.8
0.6
0.4
0.2
0.0
0.0

Relative height along the dominant stem
Figure 3.4: Dominant stem (model 3.8), represented by a dashed line, and whole-tree
(model 3.11), represented as the solid line, relative volume proﬁles as a function of relative
height along the dominant stem for the (a) minimum, (b) median, and (c) maximum
diameter at breast height (DBH) American beech (FG) trees in the dataset. The horizontal
dashed line indicates a relative volume of 0.5, the centroid of volume. The dotted vertical
line indicates the relative height of the centroid of volume for the dominant stem (DOM.c),
the vertical dashed line represents relative crown height, and the dash and dot vertical line
indicates the relative height of the centroid of volume for the whole tree (WH.c).

40

3.4 Discussion
The majority of trees in the dataset had whole-tree volume estimates with a coeﬃcient
of variation less than 10 percent. However, a few trees had a much larger coeﬃcient of
variation. Trees with a high coeﬃcient of variation appeared independent of DBH. For those
trees with a large coeﬃcient of variation, more than two paths may be necessary to decrease
the variation in the mean total whole-tree volume estimate due to a more complex branching
structure.
Dominant stem cumulative volume proﬁles required total dominant stem volume and
total height as predictor variables into the model. Total height is a measurable attribute
for a standing tree using common mensuration techniques. An estimate of total dominant
stem volume can be calculated, as in this study, using Smalian’s formula by sampling either
standing trees with a dendrometer or felling trees. Many other models in the literature
have been developed for estimation of total dominant stem volume using allometric scaling
principles (e.g., Avery and Burkhart 2002) that could be applied prior to using the cumulative
dominant stem proﬁle.
The segmented whole-tree cumulative volume proﬁles require the same predictor variables
as the model for dominant stem with additional measures of relative crown height, cumulative
volume at relative crown height, and total whole-tree volume. Relative crown height and
cumulative volume at relative crown height can be estimated on either a standing tree with
a dendrometer or felling trees. The estimate for total whole-tree volume is not as straightforward. Unlike total dominant stem volume, models using allometric scaling principles for
whole-tree volume are not as prevalent (MacFarlane 2011). MacFarlane (2011) identiﬁed
the diameter of the largest branch and crown ratio as the two best metrics for the ratio of
whole-tree to dominant stem volume. Van Deusen and Roesch (2011) oﬀer another nondestructive sampling method of diﬀerence sampling to estimate total whole-tree volume
without dividing the estimates into proﬁles. Otherwise, destructive sampling is currently

41

necessary to estimate the total whole-tree volume either by measuring the entire tree or by
random branch sampling (Gregoire et al. 1995). Any of these methods would be necessary
prior to using the segmented whole-tree cumulative volume proﬁle.
A species-composite model was shown to be suﬃcient for the cumulative dominant stem
volume proﬁle for the hardwood trees in the dataset. However, the segment of the whole-tree
model above relative crown height showed a sugar maple species-speciﬁc model resulted in
a lower AIC than the all-species model. A possible explanation could be the narrower size
range covered by the sampled maple trees having a similar branching structure compared
to the species-composite model ﬁtted to all trees across a wider size range. The conﬁdence
intervals of the ﬁxed-eﬀect parameter for the species-composite and species-speciﬁc models
do overlap. A species-composite model is also recommended for the segmented whole-tree
cumulative volume proﬁle. Zakrzewski (2011) showed diﬀerent cumulative whole-tree models
are needed for sugar maple trees with merchantable branch volume (deﬁned as branches with
diameter greater than 9 cm) compared to those without merchantable branches. All of the
trees except for three in the three smallest DBH classes had by deﬁnition merchantable
branches.
The mean centroid of dominant stem volume for all trees in the dataset was 0.222. MacFarlane (2010) found a very similar mean centroid of dominant stem volume, 0.217, for
11 common hardwood tree species in Michigan with merchantable sized branches. However, MacFarlane (2010) found the mean centroid of dominant stem volume, 0.256, for trees
without merchantable branches to be signiﬁcantly diﬀerent from trees with merchantable
branches. As the majority of the sampled trees had merchantable branches, it is expected
the mean centroid of dominant stem volume would be similar to the mean value of other
trees with merchantable branches. Both of these values for trees with merchantable branches
are lower than the 0.3 value for the centroid identiﬁed by Forslund (1982) for aspen (Populus
tremuloides Michx.) trees indicating a reduction in dominant stem volume due to branching.
Another possible diﬀerence among the datasets is the smaller trees and exclusion of forked

42

trees used by Forslund (1982).
The mean centroid of whole-tree volume for all trees in the dataset was 0.390, and
signiﬁcantly larger than the centroid of dominant stem volume. The only evident pattern
seen for both the centroid of dominant stem volume and the centroid of whole-tree volume
was between the centroid of volume and crown ratio class. The pattern of a decreasing
centroid of volume with increasing crown ratio class is indicative of branch volume dispersed
throughout the height of a tree resulting in a lower centroid.
The ﬁnal mixed-eﬀects cumulative volume models developed here may only be relevant for
hardwood trees with signiﬁcant proportions of branch volume to whole-tree volume compared
to conifers. The selected mixed-eﬀects models were all compatible with the total estimates
from Smalian’s formula for dominant stems and with random branch sampling for whole
trees.

43

CHAPTER 4
CUMULATIVE MASS AND DENSITY PROFILES FOR DOMINANT
STEMS AND WHOLE TREES

4.1 Related Biomass Modeling
4.1.1 Diameter at Breast Height Based Biomass Modeling
Whole-tree biomass equations are in great demand due to the simultaneous need to improve
estimation of forest carbon stocks and to quantify the distribution of wood mass within
trees for estimating whole-tree utilization potential (Avery and Burkhart 2002). Generalized
biomass equations have recently become more important with the desire to have consistency
at the national scale. For the U.S., Jenkins et al. (2003) compiled published biomass equations to create a nationally consistent diameter at breast height (DBH, measured at 1.37
m above ground level) based biomass equation. Several problems with this method were
summarized by the Canadian national biomass study (Lambert et al. 2005): (1) pseudodata
were used as if it was real data for parameter estimation, which generated highly correlated
observations, (2) the variances of the estimates and tests of signiﬁcance are not valid, (3)
bias in the sample selection because a study conducted in a single area was equally weighted
to a study conducted in multiple areas, and (4) selecting publications could also have a bias
in those chosen. Lambert et al. (2005) improved upon some of these drawbacks by compiling
actual data from a previous study that was designed for sampling biomass at the provincial
and regional level. However, they too had large regions of Canada with missing data from
an entire province (e.g., British Columbia) and the sampling methods were not consistent
across all provinces.
Hansen (2002) compared regional diﬀerences between total biomass models in the Eastern
U.S. used by Forest Inventory and Analysis (FIA) program of the USDA Forest Service. For

44

example, the Northeast region used a DBH only model to predict total biomass, and the
Central States and Lake States both used DBH and gross cubic foot volume to predict total
biomass. The use of diﬀerent models by region for a speciﬁc species and DBH gave varying
results to the amount of total biomass in a particular tree. These studies show the need
for consistent sampling methodology across regions in the development of national biomass
equations.

4.1.2 Component Ratio Method
In 2009, the FIA addressed the regional inconsistencies in tree biomass estimation by adopting the component ratio method (Heath et al. 2009), and Woodall et al. (2011) outlined
the methods with examples. The current FIA methods use regional volume equations for
dominant stems which are converted to mass by species-speciﬁc constant densities compiled
by Miles and Smith (2009). A component ratio adjustment factor is calculated between the
biomass derived from the regional volume equations and the dominant stem biomass estimates from Jenkins et al. (2004). The same component ratio is used to derive the estimates
of whole tree and tree components (e.g., branches and foliage).

4.1.3 Weight-ratio and Density-integral Biomass Modeling
Parresol (1999) reviewed several diﬀerent methods of biomass modeling, including the weightratio and density-integral approaches, beyond DBH-based biomass modeling with examples
for individual trees and stands of trees. The weight-ratio approach estimates the ratio of the
merchantable dominant stem mass to whole-tree mass. Many diﬀerent model forms exist for
modeling the ratio, as demonstrated by Parresol et al. (1987).
The density-integral approach to biomass modeling was ﬁrst proposed by Parresol and
Thomas (1989), which integrates a density model with a taper model over the entire height
of a tree. The speciﬁc model was applied to slash pine with predictor variables of relative
height and tree age for the density function integrated with a simple taper function requiring

45

measurements of diameter inside bark at diﬀerent relative heights and DBH. They found the
density-integral approach gave more precise estimates of biomass compared to the weightratio approach. Parresol and Thomas (1996) further examined the density-integral approach
for slash pine and willow oak for correlations among taper, volume, and mass equations.
They found the use of seemingly unrelated regression was more eﬃcient than ordinary least
squares regression. The density function employed also had predictor variables of relative
height and tree age integrated with a simple geometric taper function requiring measurements
of diameter inside bark at diﬀerent relative heights and DBH. Jordan et al. (2006) also used
seemingly unrelated regression with the density-integral approach and found a non-linear
model for density to ﬁt the best in relation to relative height for slash pine. They also used a
self-referencing density function with density at DBH as the reference observation to improve
estimation of biomass. Here, a variation of the density-integral approach is employed in this
study to take the ﬁrst derivative of a vertical volume (dominant or whole-tree) proﬁle to get
a taper model before integrating with a density proﬁle. Prior to the mass proﬁle modeling,
the necessary density proﬁles must be developed to generate the mass models.

4.2 Related Tree Density Modeling
Tree density is of interest to scientists for a number of reasons, and primarily because it is
a determinant of wood quality (Zobel and Buijtenen 1989, Shmulsky and Jones 2011). The
density of a tree is dependent upon many factors such as species, geographical location, age,
crown class, and environmental factors (U. S. D. A. Forest Service 2010). Wood density is
a key variable in tree biomass and forest carbon estimates, especially in the tropics where
large variability of wood density within an individual site exisits (Muller-Landau 2004, Chave
et al. 2009). The variation in wood density is an important factor in examining the regional
variability of aboveground biomass in the Amazon (Baker et al. 2004, Chave et al. 2006).
Wood density is generally examined in one of two ways: either the radial direction or the
longitudinal direction of a tree. The change in radial density has been studied by sampling
46

tree cores or whole-discs at DBH. Wiemann and Williamson (1989) examined the speciﬁc
gravity (sg, a unitless measurement similar to density) for some tropical and temperate tree
species. The speciﬁc gravity of the tropical trees associated as pioneer species increased from
the pith to the bark; however, the temperate species did not show a clear pattern. Woodcock
and Shier (2002; 2003) studied six temperate species in which the three species classiﬁed
as early-successional had a radial increasing speciﬁc gravity pattern, and the three latesuccessional species had a radial decreasing speciﬁc gravity pattern. They also determined
a species average speciﬁc gravity of 0.55 to be the cutoﬀ between an increasing (sg < 0.55)
and decreasing (sg > 0.55) pattern in the radial direction. Barbour et al. (1994) examined
the eﬀects of diﬀerent thinning intensities on wood density in the radial direction for jack
pine. Thinning was shown to decrease the wood density of the residual trees.
Fewer studies have examined changes in density in the longitudinal direction than the
radial direction. Tasissa and Burkhart (1998) modeled both radial and longitudinal gradients
for wood speciﬁc gravities across a large region of loblolly pine’s range. They found thinning
to have a limited eﬀect on the ring speciﬁc gravity of loblolly pine. Speciﬁc gravity was
also shown to vary by region, to decrease along the longitudinal gradient of tree height,
and to increase in the radial direction before reaching a constant level. Antony et al. (2010)
also found the same speciﬁc gravity patterns for loblolly pine as Tasissa and Burkhart (1998)
when modeling speciﬁc gravity in loblolly pine for longitudinal variation in an individual tree
across the natural range of loblolly pine. For red pine, Baker and Shottafer (1968) found
speciﬁc gravity to increase along a longitudinal gradient, but the pattern was also dependent
upon the spacing of the plantation trees. However, Wahlgren et al. (1966) studied six conifers
in Maine and found speciﬁc gravity of red pine to decrease along a longitudinal gradient.
Repola (2006) examined the vertical densities of Scots pine, Norway spruce, and birch in
Finland using a linear mixed-eﬀects model form. The density of each species was found to
be dependent upon vertical location to varying degrees. The general consensus from these
studies is that density decreases with increasing height along the dominant stem, and the

47

majority of studies examine the density of conifers.
A greater understanding of branch density is necessary for quantifying the density of the
whole tree, which presently has not been studied, for indirect estimation of whole-tree mass
and the mass of individual components of trees. A major issue in tree mass estimation is how
to determine a density value for the whole tree when diﬀerent parts of the tree have diﬀerent
densities. One indirect method of biomass estimation that is prevalent in the literature
is the density-integral approach (Parresol 1999), discussed in the previous section. Jordan
et al. (2006) examined many density proﬁles of both linear and non-linear form and found
a non-linear model for density to ﬁt the best in relation to relative height for slash pine
incorporated into the density-integral approach. A second method that will be compared
in this research will be referred to as the constant density approach. The USDA Forest
Service’s Forest Inventory and Analysis (FIA) program for example, which is responsible for
collecting and maintaining national level inventory data for forest variables, uses constant
values for wood density (or speciﬁc gravity) in their methods and models for estimating tree
biomass (Woodall et al. 2011). The method outlined uses constant values of wood density
by species from the Forest Products Laboratory (FPL) and both wood density and bark
density by species are compiled from the literature by Miles and Smith (2009).
Tree density is comprised of both wood and bark components. The separation of wood
and bark is necessary in combining an inside bark taper model with a wood density proﬁle
for biomass estimation (e.g., Jordan et al. 2006). A greater amount of research has examined
wood density compared to bark density because of diﬀerences in economic value. Martin and
Crist (1968) found the density of the bark of eastern hardwoods to be much greater than
softwoods and also quite variable within an individual species. Lamb and Marden (1968)
took bark samples at DBH and divided bark into components of inner and outer bark to
compare diﬀerent speciﬁc gravities of selected pulpwood species from Minnesota. They found
varying relationships between the two bark components by species. The speciﬁc gravity of
bark also is dependent upon the height at which the sample is taken (Koch 1970, Manwiller

48

1979). Koch (1970) found bark speciﬁc gravity to increase with height before decreasing as
heights approached the top of the tree for selected hardwood species. Manwiller (1979) also
compared dominant stem bark speciﬁc gravity with dominant stem wood, branch wood, and
branch bark. The relationships gave diﬀerent results among the species examined.
The primary objective of this chapter is to develop a new vertical proﬁle mass model
to relate whole-tree mass to dominant stem mass, which can estimate component-wise tree
attributes of dominant stem and branches. The new model is then compared to models
using the density-integral approach and a constant species-speciﬁc density throughout an
entire tree approach. Additional objectives for the density portion of this chapter are: (i) to
determine if signiﬁcant diﬀerences exist between the density proﬁle of the dominant stem of
trees and random paths proﬁling other parts of trees, (ii) to determine if diﬀerent hardwood
tree species growing in the same forest show similar patterns in their vertical density proﬁles,
both dominant and random path proﬁles, and (iii) to examine the contribution of bark to
the vertical density proﬁles for trees.

4.3 Models
The data used for density and mass model development were trees sampled from a secondgrowth maple-beech stand at Fred Russ Experimental Forest (see Ch. 2).

4.3.1 Dominant Stem Vertical Density Models
General linear regression modeling was used in developing vertical density proﬁles for dominant stems, including terms to describe the size of the tree (i.e., DBH and tree height)
and the location of measurements along the dominant stem (i.e., relative crown height and
relative height) as a third-degree polynomial function of relative height:
ρij = β0 + β1 Dj + β2 Hj + β3 RCHj + β4 rhij + β5 rhij 2 + β6 rh3 + ij ,
ij

49

(4.1)

where ρij , is basic density of stem position i for tree j in kg , Dj is tree diameter at breast
3
m

height in cm, Hj is total tree height in m, RCHj is tree relative crown height, rhij is relative
height of position i for tree j (ranging from 0 at ground level to 1 at the tip of the dominant
stem), β0 , ..., β6 are parameters to be estimated, and ij is a random error term. Total
height and relative height of the tree are computed from the total and relative length of the
dominant path through the tree.
Statistical analyses were performed in the R statistical environment (R Development
Core Team 2010). A step function with both forward and backward selection, including all
interaction terms, was used to determine the best model with the lowest Aikaike’s Information Criterion (AIC). After the ﬁnal step, the selected model was checked for statistical
signiﬁcance by an F-test followed by checking the individual parameters for statistical signiﬁcance from zero at an α of 0.05 level by a t-test, until only statistically signiﬁcant terms
remained. For some ﬁnal models, the inclusion of all interaction terms resulted in higher
order terms than a third-degree polynomial function of relative height. A multinomial categorical variable for species was added to eq. 4.1 to test for species diﬀerences as follows:
ρij = β0 + β1 T A + β2 P S + β3 CG + β4 U R + β5 AS

(4.2)
2

+ β6 Dj + β7 Hj + β8 RCHj + β9 rhij + β10 rhij + β11 rh3
ij

+ ij ,

where T A, P S, CG, U R, and AS are indicator variables with values of 0 or 1 depending
upon the species. A t-test determined the species eﬀect was statistically signiﬁcant from
zero, so that species-speciﬁc mixed-eﬀects models were developed for beech and maple trees;
species-speciﬁc models were not developed for the others because only one individual was
sampled.
Mixed-eﬀects modeling was chosen because of the hierarchical structure of the dataset
with measurements taken at the tree-level and within-tree level; density measurements taken
closer together within a single tree are more likely to be similar than distant measurements
on the same tree. The nlme package was used to ﬁt the mixed-eﬀects models (Pinheiro et al.
2011). A combination of diﬀerent random eﬀects were added to each model, and the model
50

with the lowest AIC was selected. The assumption of an unstructured covariance structure
was used for the random eﬀects.
Relative crown height (RCH) was included as a possible signiﬁcant independent variable
for the dominant stem models of all species, American beech, and sugar maple because of
the possible inﬂuence branching beginning at diﬀerent relative heights has on the density
of the dominant stem. Table 4.1 shows the summary statistics of the RCH for the three
species groups. However, the model forms for all species and American beech changed
completely with the inclusion of RCH and gave larger AIC and random error values compared
to excluding RCH from models 4.1 and 4.2. For sugar maple, RCH was found to be a nonsigniﬁcant independent variable in the model development. Thus, RCH was omitted from
all of the ﬁnal models.
Table 4.1: Relative crown height of the dominant stem (RCH) summary statistics for all
species (ALL), American beech (FG), and sugar maple (AS).
Species Mean RCH SD
Min
Max
ALL
0.356
0.105 0.107 0.508
FG
0.340
0.122 0.107 0.508
AS
0.367
0.091 0.228 0.500

4.3.2 Random Branch Path Density Models
Random branch path density proﬁles were developed to examine how diﬀerent any other
path in a tree is compared to the dominant stem density proﬁle. The destructively sampled
random branch path data for each tree was ﬁt to a density proﬁle model following the same
procedures described for the dominant stem. The base model for the random branch path
with a species categorical variable included:
ρij = β0 + β1 T A + β2 P S + β3 CG + β4 U R + β5 AS
+ β6 Dj + β7 Lj + β8 RHDj + β9 rhij + β10 rhij 2 + β11 rh3 + ij ,
ij

51

(4.3)

is similar to model 4.2 with the exception of total random path length in m, Lj , replacing
total tree height, Hj , and the tree relative height at which the random path diverges from
the dominant stem, RHDj , replacing tree relative crown height (RCHj ).
Relative height at which the random path diverged from the dominant stem (RHD) was
included as a possible signiﬁcant independent variable for the random branch path models
of all species, American beech, and sugar maple because of the possible diﬀerent densities
of the sampled branches at diﬀerent heights oﬀ of the dominant stem. Table 4.2 shows
the summary statistics of the RHD for the three species groups. However, as was the case
for the dominant path all three groups had a RHD that was found to be a non-signiﬁcant
independent variable in the model development, so that it was not included in the ﬁnal
models.
Table 4.2: Relative height at which the random path diverges from the dominant stem
(RHD) summary statistics for all species (ALL), American beech (FG), and sugar maple
(AS).
Species Mean RHD
ALL
0.702
FG
0.655
AS
0.719

SD
Min
Max
0.178 0.185 0.961
0.197 0.185 0.934
0.161 0.400 0.961

4.3.3 Bark Density Models
A subset of one tree per species per 10-cm DBH class was used to develop bark density
models. The bark density models followed the same procedures described for the multinomial
categorical species model for the dominant stem (eq. 4.2). The separation of wood and bark
into components is important because of their structural diﬀerences. Due to a small number
of bark samples per tree for the dominant stem (DOM) and the random branch path (RBP),
the observations by path were combined when developing the bark density models for the
subset of trees (Table 4.3).

52

Table 4.3: The bark sample size by species, dominant (DOM), and random branch paths
(RBP).
Path ALL
DOM 279
RBP 86

FG AS
109 97
55 18

CG
25
2

PS
17
2

TA UR
15 16
0
9

4.3.4 Dominant Stem Cumulative Mass Models
Statistical analyses were performed in the R statistical environment (R Development Core
Team 2010). A ﬁxed-eﬀects model, shown below (see eqs. 4.4 and 4.7), was ﬁrst selected
for each individual tree with parameters estimated using the nlsList function from the nlme
package (Pinheiro et al. 2011). Mixed-eﬀects modeling was then chosen because of the
hierarchical structure of the dataset with measurements taken at the tree-level and withintree level. As measurements taken closer together within a single tree are more likely to be
similar than distant measurements on the same tree. The assumption of an unstructured
covariance structure was used for the random eﬀects. The parameters from the ﬁxed-eﬀects
model were used as starting values for the mixed-eﬀects models using the nlme function in
the nlme package. The models were ﬁt to all of the species, American beech, and sugar maple
with the best-ﬁt model being selected based on the lowest Aikaike’s Information Criterion,
AIC, value. American beech and sugar maple were ﬁtted to the model individually as
multiple observations of each of these species were present. Models were also selected based
on having the smallest mean relative diﬀerence in relation to estimated total mass.
Two additional approaches for modeling the cumulative mass proﬁle were the densityintegral approach and a constant basic density approach applied to cumulative volume proﬁles. The density-integral approach combines a basic density function and taper function in
terms of relative height using integral calculus (Parresol and Thomas 1989, Parresol 1999,
Jordan et al. 2006). The integrate function in the R statistical environment was used for
solving the integrals; however, the integrals could also be solved using integration by parts.
The integrate function was chosen over integration by parts due to need of integrating the

53

ﬁfth order polynomial function of the basic density proﬁle. For the constant density approach, published species-speciﬁc densities compiled by Miles and Smith (2009) for the U.S.
were applied to cumulative volume proﬁles of each tree.
Non-linear regression modeling was used in developing the cumulative mass proﬁles for
dominant stems in terms of relative height. The base ﬁxed-eﬀects model is analogous to
modifying the relative dominant stem volume model of MacFarlane (2010), but for mass:
mDOM = MDOM ∗ (1 − (1 − rh)α1 ) + ,

(4.4)

where mDOM is the dominant stem cumulative mass in kg at a relative height (rh), MDOM
is total dominant stem mass in kg, α1 is a parameter to be estimated, and is an error term.
The second approach for modeling the dominant stem mass was to use the densityintegral approach. The basic density function, ρ(rh), and the ﬁrst derivative of a cumulative
dominant stem volume model, vDOM (rh), were used to predict mass at any relative height:
rhu

mDOM =

rhl

ρ(rh) vDOM (rh) drh,

(4.5)

where rhl is the lower limit of relative height and rhu is the upper limit of relative height.
The ﬁnal approach to modeling the dominant stem mass was to apply a species-speciﬁc
constant basic density, ρc , throughout the entire dominant stem to a cumulative dominant
stem model, vDOM (rh):
mDOM = ρc ∗ vDOM (rh),

(4.6)

where ρc is the species-speciﬁc densities compiled by Miles and Smith (2009).
4.3.5 Whole-tree Cumulative Mass Models
The whole-tree cumulative mass model uses the modiﬁed relative dominant stem mass model
(eq. 4.4) for the portion below relative crown height with the addition of a modiﬁed mass
model where the intercept is shifted to relative crown height for the portion above relative
crown height:
mAC = MRCH + βM ∗ (1 − (1 − (rh − RCH))α2 ) + ,
54

(4.7)

where mAC is the whole-tree cumulative mass, in kg, above relative crown height at a relative
height (rh), MRCH is total tree mass at relative crown height (RCH) in kg, βM is a either
a ﬁxed parameter or an estimated parameter, α2 is a parameter to be estimated, and is an
error term. In the case where βM is ﬁxed, the new model has the same form as model 4.7
setting
βM =

MW T − MRCH
,
1 − RCH α2

(4.8)

where MW T is the total whole-tree mass in kg.
Models 4.4 and 4.7 or models 4.4 and 4.8 are combined to get the whole-tree cumulative
mass proﬁle as
mW T = I1 mDOM + I2 mAC ,

(4.9)

where mW T is the whole-tree cumulative mass in kg at a relative height, I1 is an indicator
variable with a value of 1 when relative height is below relative crown height and a value of
0 when relative height is above relative crown height, and I2 is an indicator variable with
values of 0 when relative height is below relative crown height and 1 when relative height is
above relative crown height.
Similar to the dominant stem, the second approach to modeling whole-tree mass was to
use the density-integral approach. Due to the segmented model of whole-tree mass proﬁle
(eq. 4.9), two integrals are needed to estimate whole-tree mass for relative heights above
relative crown height:
RCH

mW T =

rhl

rhu

ρ(rh) vDOM (rh) drh +

RCH

ρ(rh) vAC (rh) drh,

(4.10)

where rhl is the lower limit of relative height and rhu is the upper limit of relative height.
The dominant stem basic density proﬁle was used for both integrals for whole-tree mass
estimation as it is a more conservative estimate, and the whole tree above relative crown
height is a mixture of dominant stem and branches.
The ﬁnal approach to modeling whole-tree mass was to apply a species-speciﬁc constant basic density, ρc , throughout the entire tree to a cumulative whole-tree volume model,
55

vW T (rh):
mW T = ρc ∗ vW T (rh).

(4.11)

4.4 Results
4.4.1 Dominant Stem Vertical Density Proﬁles
The ﬁnal best-ﬁt all-species and species-speciﬁc dominant stem models were as follows:
All:
ρij = β0 + u0j + β1 T A + β2 P S + β3 CG + β4 U R + β5 AS
+ β6 rh2 + (β7 + u1j )rh3 + β8 rh4 + β9 rh5
ij
ij
ij
ij

(4.12)

+ β10 Dj rh2 + ij ,
ij
American beech:
ρij = β0 + u0j + (β1 + u1j )rh2 + β2 rh3 + β3 rh4 + β4 rh5
ij
ij
ij
ij
+ β5 Dj rh3
ij

(4.13)

+ ij ,

Sugar maple:
ρij = u0j + β1 Dj + β2 Hj + (β3 + u1j )rhij + (β4 + u2j )rh2
ij
+ β5 rh3
ij

(4.14)

+ β6 Dj Hj + ij ,

where β0 , ..., β10 are the ﬁxed eﬀects parameters to be estimated, and u0j , ..., u2j are the
random eﬀects parameters to be estimated for tree j. American beech and sugar maple
had multiple sample trees so a species-speciﬁc model was ﬁt to each of them. For the other
four species, the all-species model was applied as only one individual was sampled for each
species.
Dominant stem vertical density proﬁles for all six species varied with relative height along
the stem. Table 4.4 summarizes the best-ﬁt models for the dominant stem of each species.
Each species examined had a complex vertical density proﬁle (Fig. 4.1). American beech,
pignut hickory, and black cherry had a decreasing density from the base of the tree until
56

an approximate relative height of 0.3, density then increased back to its previous density at
the base of the tree, and then decreased once again before increasing one ﬁnal time prior to
reaching the tip of the stem. Sugar maple, basswood, and slippery elm also had a decreasing
density from the base of the tree until an approximate relative height of 0.3, and density
then began to increase once more before decreasing again towards the tip of the stem. In
comparison to the reported species average given by the Forest Products Laboratory (FPL)
in Madison, WI (U. S. D. A. Forest Service 2010), these American beech trees were nearly 50
kg/m3 denser at its maximum than the FPL average and denser for the entire vertical proﬁle.
These sugar maple, basswood, black cherry, and slippery elm had densities at the base of
the tree that began near the reported FPL species average, and showed ﬂuctuating patterns
with increasing height. However, the one hickory tree examined had a density ﬂuctuating
nearly 100 kg/m3 below the reported FPL species average.

57

Table 4.4: Parameter estimates of models 4.12 - 4.14 for the vertical proﬁle of the basic
density of the dominant stem for the all-species (ALL), American beech (FG), and sugar
maple (AS) mixed-eﬀects models. For the ﬁxed parameters, the standard error of estimates
is given in parantheses.
Parameter
β0
β1
β2
β3
β4
β5
β6
β7
β8
β9
β10
var(u0j )
var(u1j )
var(u2j )
cov(u0j , u1j )
cov(u0j , u2j )
cov(u1j , u2j )
ij

AIC

ALL Estimate (4.12)
583.179 (8.025)
-273.345 (24.221)
-105.364 (24.508)
-37.450 (23.733)
-108.073 (24.227)
-36.164 (8.960)
-1293.481 (270.449)
5739.937 (1035.995)
-7837.064 (1370.802)
3425.819 (603.524)
-1.015 (0.515)
668.328
4346.393

FG Estimate (4.13)
602.280 (11.215)
-1885.849 (424.101)
7931.054 (1621.901)
-10968.578 (2127.776)
4988.697 (928.895)
-1.543 (0.772)

-927.167

-1613.191

34.672
9022.73

38.925
4749.19

961.666
5413.880

58

AS Estimate (4.14)
12.053 (1.422)
19.296 (1.059)
-350.696 (88.638)
1153.375 (149.937)
-849.595 (85.583)
-0.424 (0.043)

2975.325
51888.220
53832.660
-9505.240
7922.540
-51583.07
24.278
3142.52

700

700

600

600

500

500

400

FG

Density (kg m3)

0.0

400
0.2

0.4

0.6

0.8

1.0

AS
0.0

700

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

300

500

0.4

400

600

0.2

200

400

CG
0.0

100
0.2

0.4

0.6

0.8

1.0

TA
0.0

600

600

500

500

400

400

300

PS
0.0

300
0.2

0.4

0.6

0.8

1.0

UR
0.0

Relative height
Figure 4.1: The mixed-eﬀects of the dominant stem vertical density proﬁle for each species
according to models 4.12 - 4.14. For the median sized American beech (FG) tree with D =
42.9 cm, and the median sized sugar maple (AS) tree with D = 42.8 cm and H = 34.0 cm
(models 4.13 and 4.14, respectively). American basswood (TA), black cherry (PS), pignut
hickory (CG), and slippery elm (UR) were all ﬁt using the all-species model (model 4.12)
as there was only one individual of each in the sample. The measured tree densities are
shown as triangles. The horizontal dotted line represents the wood density for each species
as reported by the Forest Products Laboratory in Madison, WI (U. S. D. A. Forest Service
2010).

59

4.4.2 Random Branch Path Density Proﬁles
The ﬁnal best-ﬁt all-species and species-speciﬁc models for the random branch proﬁles are
as follows:
All:
ρij = β0 + u0j + β1 T A + β2 P S + β3 CG + β4 U R + β5 AS
+ (β6 + u1j )rhij + β7 rh2 + (β8 + u2j )rh3 + β9 rh4
ij
ij
ij

(4.15)

+ β10 rh5 + β11 Lj rh3 + ij ,
ij
ij
American beech:
ρij = β0 + u0j + β1 rh2 + β2 rh3 + β3 rh4 + β4 rh5
ij
ij
ij
ij
+ β5 Lj rh3
ij

+ β6 Lj rh2
ij

(4.16)

+ ij ,

Sugar maple:
ρij = u0j + β1 Dj + β2 Lj + (β3 + u1j )rhij
+ (β4 + u2j )rh2 + β5 rh3 + β6 Dj Lj + β7 Lj rh3
ij
ij
ij

(4.17)

+ β8 Lj rh2 + β9 Lj rhij + ij ,
ij
where Lj is the total random branch path length, β0 , ..., β11 are the ﬁxed eﬀects parameters
to be estimated, and u0j , ..., u2j are the random eﬀects parameters to be estimated for tree
j. Table 4.5 summarizes the best-ﬁt models for the random branch path density of each
species. The density proﬁles for all six species derived from random branch sampling also
varied with relative height, indicating a non-constant and non-linear density proﬁle for the
tree.

60

Table 4.5: Parameter estimates of models 4.15 - 4.17 for the basic density proﬁle of the
random branch path for the all-species (ALL), American beech (FG), and sugar maple
(AS) mixed-eﬀects models. For the ﬁxed parameters, the standard error of estimates is
given in parantheses.
Parameter
β0
β1
β2
β3
β4
β5
β6
β7
β8
β9
β10
β11
var(u0j )
var(u1j )
var(u2j )
cov(u0j , u1j )
cov(u0j , u2j )
cov(u1j , u2j )
ij

AIC

ALL Estimate (4.15)
558.853 (16.698)
-281.124 (20.990)
-125.705 (22.370)
-47.121 (20.819)
-98.433 (20.768)
-43.537 (7.704)
628.361 (252.462)
-5307.591 (1371.455)
15948.619 (3146.769)
-19142.133 (3195.509)
8055.966 (1186.348)
-4.020 (0.995)
1860.944
18270.89
13815.23
-5049.686
4010.719
-15490.43
34.687
8503.868

FG Estimate (4.16)
AS Estimate (4.17)
609.963 (8.974)
-3064.950 (464.381)
8.552 (2.641)
12028.253 (1639.495) 20.658 (1.099)
-16523.545 (2113.572) 1308.307 (550.599)
7773.176 (911.859)
-2723.536 (1093.127)
-19.917 (5.895)
1689.574 (648.788)
13.817 (5.866)
-0.332 (0.101)
-79.676 (25.019)
126.844 (42.108)
-56.874 (21.199)

221.624

37.911
4288.547

2595.301
15939.370
13449.170
-5544.173
3940.647
-13821.51
27.861
3116.386

4.4.3 Comparative Dominant and Random Branch Path Density Proﬁles
Separate density models for the dominant stem and random branch path were ﬁt to examine if
statistical diﬀerences exist between the dominant stem and random branch path. The density
proﬁles were developed and compared based upon relative height or relative path length.
Dominant stem and random branch path models were considered statistically diﬀerent if
either model forms were diﬀerent or at least one parameter’s 95 percent conﬁdence interval
did not overlap (Rubin and MacFarlane 2008). Comparing the best-ﬁt model forms of models
4.12 - 4.14 to models 4.15 - 4.17 by respective species, the dominant and random branch
path densities are statistically diﬀerent from one another due to diﬀerent model forms.
Figure 4.2 shows the comparison between the vertical density of the dominant stem and

61

the random branch path for each species. Each species shows similar varying patterns for
the random branch path as the dominant stem because a majority of each random branch
path follows the dominant stem until branching is encountered, as the mean RHD for all
sampled species was 0.702. Greater densities occur for the random branch path near the
top than for the dominant stems for all species except sugar maple. This shows that the
vertical density proﬁle of the dominant path is not representative of the whole tree because
of signiﬁcant diﬀerences in the density of branches. All of the species random branch path
density proﬁles are complex and non-constant, as was the case for dominant stems, compared
to the reported species average of the FPL.
One possible explanation for the diﬀerence of vertical densities near the top of the proﬁles
was the relative height at which the dominant path and the random branch path diverged
from one another (RHD); however, RHD was not a signiﬁcant variable in the ﬁnal best-ﬁt
random branch path density models. Figure 4.3 shows three American beech trees that
represent (a) the minimum RHD, (b) a RHD near the middle of a tree, and (c) a RHD
near the top of a tree. All three trees show the divergence between the dominant stem
and random branch path models to precede the RHD. The diﬀerence between the densities
continues to the location of the RHD, but the greatest diﬀerence between densities of the
two paths occurs closer to the top of each relative path length. The signiﬁcance of the higher
order relative height terms in each model accounts for the diﬀerences between the dominant
stem and the random branch path.

62

700

700

600

600

500

500

400

FG

Density (kg m3)

0.0

400
0.2

0.4

0.6

0.8

1.0

AS
0.0

700

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

300

500

0.4

400

600

0.2

200

400

CG
0.0

100
0.2

0.4

0.6

0.8

1.0

TA
0.0

600

600

500

500

400

400

300

PS
0.0

300
0.2

0.4

0.6

0.8

1.0

UR
0.0

Relative height or path length
Figure 4.2: The mixed-eﬀects vertical density proﬁle for the dominant stem of each species
according to models 4.12 - 4.14, shown as solid lines, versus relative height. The thicker
dashed lines are the mixed-eﬀects density proﬁle for the random branch path of each species
according to models 4.15 - 4.17 versus relative path length. For the median sized American
beech (FG) tree with D = 42.9 cm, and the median sized sugar maple (AS) tree with D =
42.8 cm and H = 34.0 cm (models 4.13, 4.16 and 4.14, 4.17, respectively). American
basswood (TA), black cherry (PS), pignut hickory (CG), and slippery elm (UR) were all ﬁt
using the all-species models 4.12 and 4.15 as there was only one individual of each in the
sample. The horizontal dotted line represents the wood density for each species as given by
the Forest Products Laboratory in Madison, WI (U. S. D. A. Forest Service 2010).

63

700
650
600
550
a

500

Density (kg m3)

0.0

0.2

0.4

0.6

0.8

1.0

700
650
600
550
b

500
0.0

0.2

0.4

0.6

0.8

1.0

700
650
600
550
c

500
0.0

0.2

0.4

0.6

0.8

1.0

Relative height or path length
Figure 4.3: The mixed eﬀects of the vertical density proﬁle for the dominant stem (solid
line) and the random branch path proﬁle (thicker dashed line) for three American beech
trees versus relative height or path length. (a) Shows an American beech tree with
diameter at breast height (DBH) of 50.1 cm, relative height at which the random path
diverges from the dominant stem (RHD) equal to 0.185. (b) Shows an American beech tree
with DBH of 47.1 cm and RHD equal to 0.544. (c) Shows an American beech tree with
DBH of 42.9 cm and RHD equal to 0.888. The horizontal dotted line represents the wood
density for American beech of 560 kg/m3 as reported by the Forest Products Laboratory in
Madison, WI (U. S. D. A. Forest Service 2010).

64

4.4.4 Bark Vertical Density Proﬁles
The ﬁnal best-ﬁt all-species and species-speciﬁc models for bark density are as follows:
All:
ρij = β0 + β1 T A + β2 P S + β3 CG + β4 U R + β5 AS
+ β6 rh3
ij

+ β7 Hj rhij + u0j + u1j rh3
ij

(4.18)

+ ij ,

American beech:
ρij = β0 + β1 rh3 + β2 rh5 + β3 Dj rh2 + β4 Dj rh3
ij
ij
ij
ij
+ u0j + u1j rh3
ij

(4.19)

+ ij ,

Sugar maple:
ρij = β0 + β1 rh2 + β2 rh3 + u0j + u1j rh3 + ij ,
ij
ij
ij

(4.20)

where β0 , ..., β7 are the ﬁxed eﬀects parameters to be estimated, and u0j , u1j are the random
eﬀects parameters to be estimated for tree j. Table 4.6 summarizes the best-ﬁt models for
bark vertical density of each species.

65

Table 4.6: Parameter estimates of models 4.18 - 4.20 for the vertical proﬁle of the bark
basic density of the combined dominant and random branch path for the all-species (ALL),
American beech (FG) and sugar maple (AS) mixed-eﬀects models. For the ﬁxed
parameters, the standard error of estimates is given in parantheses.
Parameter
ALL Estimate (4.18)
β0
689.653 (16.283)
β1
-247.682 (41.389)
β2
-115.497 (41.254)
β3
-129.827 (40.949)
β4
-268.368 (41.005)
β5
-6.058 (22.996)
β6
-160.196 (24.654)
β7
3.371 (0.562)
var(u0j )
1371.970
var(u1j )
4551.853
cov(u0j , u1j ) -162.435
29.431
ij
AIC
3547.357

FG Estimate (4.19) AS Estimate (4.20)
695.107 (20.006)
679.016 (13.375)
-446.841 (110.959)
401.756 (68.538)
403.666 (122.688)
-510.651 (76.923)
8.245 (1.744)
-7.085 (2.136)

2242.472
2152.330
1195.135
24.206
1531.046

649.081
2238.560
-206.125
28.973
1108.966

For the subset of trees in which bark was separated from wood, the bark vertical density
proﬁles for all six species varied with location along relative height. The bark vertical
densities for American beech, sugar maple, black cherry, and slippery elm were greater along
the entire vertical proﬁle than those compiled by Miles and Smith (2009) and used by the
FIA (Fig. 4.4). Pignut hickory and basswood were less than the species average compiled
by Miles and Smith (2009) for nearly the entire vertical proﬁle. However, Harkin and Rowe
(1971) reports average values of bark speciﬁc gravity (using oven-dry mass and oven-dry
volume) for American beech and sugar maple that are very similar to the values of the
bark density proﬁles. Five of the six species showed a gradual increase in bark density with
increasing relative height followed by a drastic decline in bark density approaching the top of
the path, with the exception of American beech. The beech bark density actually increased
approaching the top of the path. The top of the bark proﬁles have been extrapolated
beyond the last measured relative height, represented as dashed lines in Fig. 4.4, which can
be problematic when ﬁtted with a polynomial function. Extrapolation does not appear to
be an issue for American beech, sugar maple, pignut hickory, and slippery elm as the last

66

measured relative height locations are greater than 0.9. However, some caution is necessary
for black cherry and basswood with the last measured density at a relative height near 0.8.
The vertical bark density proﬁle is greater than the vertical tree density proﬁle for the
dominant stem for the majority of the species examined (Fig. 4.5). American beech, sugar
maple, and black cherry have bark densities nearly 100 kg/m3 greater than tree densities
along the entire proﬁle, except near the tip of the stem for sugar maple. The same is true for
pignut hickory and basswood for most of the proﬁle except near the tip the bark density is
less than tree density. Slippery elm is the only one with bark less dense than the tree density
for nearly the entire proﬁle, which was also seen in the reported species average densities for
elm.

67

800

800

700

700

600

600

500

500
FG

400

Density (kg m3)

0.0

0.2

0.4

0.6

0.8

AS

400

1.0

0.0

700

0.2

0.4

0.6

0.8

1.0

500
400

600

300
500

200
CG

400
0.0

0.2

0.4

0.6

0.8

TA

100

1.0

0.0

600

0.2

0.4

0.6

0.8

1.0

500
400

500
300
PS

400
0.0

0.2

0.4

0.6

0.8

UR

200

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Relative height
Figure 4.4: The mixed eﬀects of the vertical bark density proﬁle for the combined
dominant stem and random branch paths of each species according to models 4.18 - 4.20,
shown as solid lines. The thicker dashed line represents the extrapolation of the models
beyond the last measured relative height. For the median sized American beech (FG) tree
with D = 50.1 cm, and the median sized sugar maple (AS) tree with D = 48.1 cm (models
4.19 and 4.20, respectively). American basswood (TA), black cherry (PS), pignut hickory
(CG), and slippery elm (UR) were all ﬁt using the all-species model 4.18 as there was only
one individual of each in the sample. The horizontal dotted line represents the bark density
for each species as compiled by the USDA Forest Service (Miles and Smith 2009).

68

800

700
650

700

600
600

550
FG

500

Density (kg m3)

0.0

0.2

0.4

0.6

0.8

AS

500

1.0

0.0

600

0.2

0.4

0.6

0.8

1.0

500
400

550

300
500

200
CG

450
0.0

0.2

0.4

0.6

0.8

TA

100

1.0

0.0

600

0.2

0.4

0.6

0.8

1.0

500

550

450

500
400

450
PS

400
0.0

0.2

0.4

0.6

0.8

UR

350

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Relative height
Figure 4.5: The mixed eﬀects of the vertical bark density proﬁle for the combined
dominant stem and random branch paths of each species according to models 4.18 - 4.20,
shown as solid lines, with dashed lines representing the extrapolation of the models beyond
the last measured relative height. For the median sized American beech (FG) tree with D
= 50.1 cm, and the median sized sugar maple (AS) tree with D = 48.1 cm (models 4.19
and 4.20, respectively). American basswood (TA), black cherry (PS), pignut hickory (CG),
and slippery elm (UR) were all ﬁt using the all-species model 4.18 as there was only one
individual of each in the sample. The dotted lines represent the tree density for each
species from models 4.12 - 4.14.

69

4.4.5 Total Mass Estimates
Table 4.7 shows the summary statistics for the total mass of dominant stems and whole trees
by species and for all trees in the dataset. The total mass of both dominant stems and whole
trees show an increasing pattern for larger DBH for all species, American beech, and sugar
maple (Fig. 4.6).
Table 4.7: Species code (spp), number of trees (n), mean dominant stem mass (MDOM ),
mean total mass (MW T ) with standard deviation (SD), minimum (min), and maximum
(max) values by species.
spp
FG
AS
CG
PS
TA
UR
ALL

n
15
13
1
1
1
1
32

Mean MDOM (kg) [SD, min, max] Mean MW T (kg) [SD, min, max]
2140 [1327, 81, 4552]
5349 [5801, 131, 24354]
1196 [728, 410, 2536]
2047 [1395, 557, 5043]
2772 [NA, NA, NA]
4492 [NA, NA, NA]
170 [NA, NA, NA]
229 [NA, NA, NA]
455 [NA, NA, NA]
620 [NA, NA, NA]
2973 [NA, NA, NA]
6541 [NA, NA, NA]
1688 [1189, 81, 4552]
3711 [4403, 131, 24354]

70

25000
WT
DOM

20000
15000
10000
5000

ALL

0

Total Mass (kg)

0

20

40

60

80

100

25000
20000
15000
10000
5000
FG

0
0

20

40

60

80

100

5000
4000
3000
2000
1000

AS

0
0

20

40

60

80

100

DBH (cm)
Figure 4.6: The total mass, in kg, of sampled trees for the dominant stem (DOM) and
whole tree (WT) versus diameter at breast height (DBH), in cm, for all trees (ALL),
American beech (FG), and sugar maple (AS).

71

4.4.6 Cumulative Mass Proﬁles
Dominant Stem Mass Proﬁles
The ﬁnal model, below, for the dominant stem mass added a random eﬀect to model 4.4
to create a mixed-eﬀects model. The ﬁnal cumulative mass of the dominant stem model
applied to all species, American beech, and sugar maple is
α +u
mDOMij = MDOMj ∗ (1 − (1 − rhij ) 1 1j ) + ij ,

(4.21)

where mDOMij is the cumulative mass of stem position i for tree j in kg, MDOMj is total
dominant stem mass in kg, rhij is relative height of position i for tree j, α1 is a ﬁxed eﬀect
parameter to be estimated, u1j is a random eﬀect parameter to be estimated, and ij is a
random error term. Table 4.8 gives the parameter estimates for model 4.21. The speciesspeciﬁc models are selected over the all-species model as the best models because they have
lower AIC values. The total dominant stem mass given by the model is also compatible with
the ﬁnal observation of total dominant stem mass, as the mean relative diﬀerence is equal
to zero.
Table 4.8: Parameter estimates of model 4.21 for the cumulative mass of the dominant
stem for the all-species (ALL), American beech (FG), and sugar maple (AS) mixed-eﬀects
models. For the ﬁxed parameters, the standard error of estimates is given in parantheses.
Parameter
α1
var(u1j )
ij

AIC
Mean Relative Diﬀerence

ALL Estimate FG Estimate
2.775 (0.049)
2.865 (0.068)
0.069
0.064
25.605
29.532
18359.7
9821.4
0
0

AS Estimate
2.722 (0.072)
0.066
15.694
6088.0
0

The ﬁnal model for the density-integral approach to modeling the dominant stem mass is
similar to model 4.5 with the addition of subscripts for measurements taken at stem position
i for tree j:
rhu

mDOMij =

rhl

ρij vDOM

ij

72

drh,

(4.22)

where rhl is the lower limit of relative height and rhu is the upper limit of relative height.
The density function for the dominant stem is a species-speciﬁc function, and the taper
function is a species-composite function.
The ﬁnal constant density approach model also applies a species-speciﬁc constant density
and a species-composite cumulative volume function similar to model 4.6 with the addition
of subscripts for measurements taken at stem position i for tree j:
mDOMij = ρc ∗ vDOMij .

(4.23)

Figures 4.7 and 4.8 show the dominant stem cumulative mass for the (a) minimum, (b)
median, and (c) maximum DBH of American beech and sugar maple trees, respectively.
For each tree, the proﬁles are shown based upon the three modeling approaches of (1) nonlinear mixed-eﬀects (nlme), (2) density-integral (integ), and (3) constant density (const).
For both species the non-linear mixed-eﬀects and density-integral approaches gave similar
results throughout the proﬁle; however, the non-linear mixed-eﬀects approach assured that
the total dominant stem mass observed was equal to the total dominant stem mass predicted.
For the American beech trees, the constant density approach underestimated the observed
dominant stem mass throughout the proﬁles. The opposite was true for the sugar maple
trees with the constant density approach overestimating the observed dominant stem mass
throughout the proﬁles.

73

a

80
60
40

nlme
integ
const

20
0

Cumulative
Mass (kg)

0.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

b

1000
800
600
400
200
0
0.0

c

4000
3000
2000
1000
0
0.0

Relative height along dominant stem
Figure 4.7: Cumulative mass, in kg, proﬁles for the dominant stem modeled by: (1) the
non-linear mixed-eﬀects approach (model 4.21, nlme), (2) the density-integral approach
(model 4.22, integ), and (3) the constant density approach (model 4.23, const) as a
function of relative height along the dominant stem for the (a) minimum, (b) median, and
(c) maximum diameter at breast height (DBH) American beech (FG) trees in the dataset.
The vertical dashed line represents relative crown height.

74

a

400
300
200

nlme
integ
const

100
0

Cumulative
Mass (kg)

0.0

1200
1000
800
600
400
200
0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

b

0.0

2500
c

2000
1500
1000
500
0
0.0

Relative height along dominant stem
Figure 4.8: Cumulative mass, in kg, proﬁles for the dominant stem modeled by: (1) the
non-linear mixed-eﬀects approach(model 4.21, nlme), (2) the density-integral approach
(model 4.22, integ), and (3) the constant density approach (model 4.23, const) as a
function of relative height along the dominant stem for the (a) minimum, (b) median, and
(c) maximum diameter at breast height (DBH) sugar maple (AS) trees in the dataset. The
vertical dashed line represents relative crown height.

75

Table 4.9 shows the summary statistics for residuals of the three methods of modeling
dominant stem mass proﬁles for all trees, American beech, and sugar maple. The magnitude
of the mean residual was lowest for the density-integral approach for all trees, sugar maple
only, and nearly equal to the non-linear mixed-eﬀects approach for American beech only.
The constant density approach had a much larger magnitude for the mean residual for all
trees, American beech, and sugar maple compared to the other two methods.
Table 4.9: Model comparison of the residuals among the cumulative dominant stem mass
proﬁles: (1) non-linear mixed-eﬀects (nlme) model 4.21, (2) density-integral (integ) model
4.22, and (3) the constant density (const) model 4.23 for all (ALL), American beech (FG),
and sugar maple (AS) trees.
Statistic
nlme
integ
const
ALL Trees
N
1957
1957
1957
Mean Residual 3.89
3.38
-7.19
SD
25.12
32.10
138.71
Min Residual
-139.50 -155.87 -630.07
Max Residual 419.72 432.57 461.81
FG Trees
N
Mean Residual
SD
Min Residual
Max Residual

1016
4.49
29.00
-139.50
419.72

AS Trees
N
Mean Residual
SD
Min Residual
Max Residual

723
723
723
2.46
-1.17
-60.11
15.37
16.80
53.10
-101.70 -138.84 -188.17
127.52 113.01 105.10

1016
1016
4.51
70.16
38.95
72.50
-155.87 -65.52
432.57 461.81

Whole-tree Mass Proﬁles
The ﬁnal model, below, for whole-tree mass added random eﬀects to model 4.7 to create a
mixed-eﬀects model. Two potential ﬁnal mixed-eﬀects models for the portion of the whole-

76

tree model above relative crown height are given below:
α +u
mACij = MRCHj + (βM + u3j ) ∗ (1 − (1 − (rhij − RCHj )) 2 2j ) + ij ,

(4.24)

where mACij is the whole-tree cumulative mass above relative crown height of stem position
i in tree j in kg, MRCHj is total tree mass at relative crown height (RCH) in kg, βM and
α2 are ﬁxed eﬀect parameters to be estimated, u2j and u3j are random eﬀect parameters
to be estimated, and ij is a random error term. In the case where βM is ﬁxed for tree j,
the new model has the same form as model 4.24, except only one random eﬀect parameter
(u2j ), with
βM j =

MW Tj − MRCHj

α +u
1 − RCHj 2 2j
where MW Tj is the total whole-tree mass for tree j in kg.

,

(4.25)

The ﬁnal whole-tree cumulative mass proﬁle (similar to eq. 4.9) now uses models 4.21
and 4.24 or models 4.21 and 4.25 for the segmented model:
mW Tij = I1 mDOMij + I2 mACij ,

(4.26)

where mW Tij is the whole-tree cumulative mass of stem position i in tree j in kg, I1 is an
indicator variable with a value of 1 when relative height is below relative crown height and a
value of 0 when relative height is above relative crown height, and I2 is an indicator variable
with values of 0 when relative height is below relative crown height and 1 when relative
height is above relative crown height. Tables 4.10 and 4.11 give the parameter estimates for
models 4.24 and 4.25.
Model 4.25 is chosen as the best-ﬁt ﬁnal model of the two for the whole-tree above relative
crown height portion of the segmented mass model as it has a lower AIC of 22219.50 for the
all-species model compared to an AIC of 22505.42 for the all-species model 4.24. The total
whole-tree mass for model 4.25 is also compatible with the total whole-tree mass estimate
given by random branch sampling as the mean relative diﬀerence is zero. The species-speciﬁc
models for American beech and sugar maple for model 4.25 are selected as the best models
over the all-species model based on having lower AIC values.
77

Table 4.10: Parameter estimates of model 4.24 for cumulative whole-tree mass above
relative crown height for the all-species (ALL), American beech (FG), and sugar maple
(AS) mixed-eﬀects models. For the ﬁxed parameters, the standard error of estimates is
given in parantheses.
Parameter
α2
β
var(u2j )
var(u3j )
cov(u2j , u3j )

ALL Estimate
FG Estimate
4.285 (0.298)
4.253 (0.399)
2590.80 (663.23) 3943.47 (1286.95)
2.209
2.044
14058457
24787417
256.340
476.850
140.281
185.157
ij
AIC
22505.42
12313.63
Mean Relative Diﬀerence 0.520
0.579

AS Estimate
4.306 (0.410)
1255.66 (284.75)
2.011
1050508
-168.604
51.497
6966.18
0.703

Table 4.11: Parameter estimates of model 4.25 for cumulative whole-tree mass above
relative crown height for the all-species (ALL), American beech (FG), and sugar maple
(AS) mixed-eﬀects models. For the ﬁxed parameters, the standard error of estimates is
given in parantheses.
Parameter
α2
var(u2j )
ij

AIC
Mean Relative Diﬀerence

ALL Estimate FG Estimate AS Estimate
4.430 (0.283)
4.410 (0.399) 4.415 (0.377)
2.065
2.086
1.718
142.080
186.862
54.809
22219.50
12170.65
6916.13
0
0
0

The ﬁnal density-integral approach model to the whole-tree mass proﬁle is similar to
model 4.10 with the addition of subscripts for measurements taken at stem position i for
tree j:
mW Tij =

RCHj
rhl

rhu

ρij vDOM

ij

drh +
RCHj

ρij vAC

ij

drh,

(4.27)

where rhl is the lower limit of relative height and rhu is the upper limit of relative height.
The species-speciﬁc dominant stem density function is used for both integrals, and the taper
function for both integrals is a species-composite function.
The ﬁnal model for the constant density approach also applies a species-speciﬁc constant
basic density and a species-composite cumulative volume function similar to model 4.11 with

78

the addition of subscripts for measurements taken at stem position i for tree j:
mW Tij = ρc ∗ vW Tij .

(4.28)

Figures 4.9 and 4.10 show the dominant stem and whole-tree cumulative mass proﬁle for
the (a) minimum, (b) median, and (c) maximum DBH trees of American beech and sugar
maple, respectively. For American beech, the total whole-tree mass increases by a factor of
ten in each successive proﬁle, and the relative crown height also decreases with increasing
DBH. A noticeable increase in branch mass, which is the region between the whole-tree and
dominant stem cumulative mass proﬁles, is also present in each successive proﬁle. Comparing
the three methods for whole-tree mass estimation above relative crown height, the non-linear
mixed-eﬀects model ﬁts all three trees well. Both the density-integral and constant density
approaches show worse ﬁts to the observed data with each successive proﬁle. The sugar maple
proﬁles show similar patterns to the American beech with total whole-tree mass increasing
by a factor of three in each successive proﬁle. However, the relative crown height appears to
increase before decreasing with increasing DBH, and the branch mass also decreases before
increasing with increasing DBH. The three methods for whole-tree estimation above relative
crown height appear to ﬁt the three proﬁles well for sugar maple with slight over-prediction
and under-prediction for the density-integral and constant density approaches.

79

150

a

100
nlme
integ
const

50
0

Cumulative
Mass (kg)

0.0

2000

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

b

1500
1000
500
0
0.0

25000
c

20000
15000
10000
5000
0
0.0

Relative height along the dominant stem
Figure 4.9: Cumulative mass, in kg, proﬁles for the dominant stem (model 4.21),
represented by a solid line and open triangles above relative crown height, and whole tree
(model 4.26, nlme), represented by a solid line with ﬁlled-in triangles, as a function of
relative height along the dominant stem for the (a) minimum, (b) median, and (c)
maximum diameter at breast height (DBH) American beech (FG) trees in the dataset.
Above relative crown height the density-integral (model 4.27, integ) and constant density
(model 4.28, const) models are shown with dotted and dashed lines, respectively. The
vertical dashed line represents relative crown height.

80

600
500
400
300
200
100
0

a

nlme
integ
const

Cumulative
Mass (kg)

0.0

1500

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

b

1000
500
0
0.0

5000
4000
3000
2000
1000
0

c

0.0

Relative height along the dominant stem
Figure 4.10: Cumulative mass, in kg, proﬁles for the dominant stem (model 4.21),
represented by a solid line and open triangles above relative crown height, and whole tree
(model 4.26, nlme), represented by a the solid line with ﬁlled-in triangles, as a function of
relative height along the dominant stem for the (a) minimum, (b) median, and (c)
maximum diameter at breast height (DBH) sugar maple (AS) trees in the dataset. Above
relative crown height the density-integral (model 4.27, integ) and constant density (model
4.28, const) models are shown with dotted and dashed lines, respectively. The vertical
dashed line represents relative crown height.

81

Table 4.12 shows the summary statistics for residuals of the three methods of modeling
whole-tree mass proﬁles for all trees, American beech, and sugar maple. The magnitude of
the mean residual was lowest for the non-linear mixed-eﬀects approach for all trees, American
beech, and sugar maple. The density-integral and constant density approaches had much
larger magnitudes for the mean residual for all trees, American beech, and sugar maple.
Table 4.12: Model comparison of the residuals among the cumulative whole-tree mass
proﬁles above relative crown height: (1) non-linear mixed-eﬀects (nlme) model 4.25, (2)
density-integral (integ) model 4.27, and (3) the constant density (const) model 4.28 for all
(ALL), American beech (FG), and sugar maple (AS) trees.
Statistic
nlme
integ
ALL Trees
N
1735
1735
Mean Residual 8.21
322.66
SD
140.85
996.12
Min Residual
-2924.47 -795.95
Max Residual 2150.16 6471.32

const
1735
311.67
1038.47
-1255.90
6406.94

FG Trees
N
Mean Residual
SD
Min Residual
Max Residual

911
11.64
185.25
-2924.47
2150.16

911
911
635.94 743.02
1292.55 1261.23
-795.95 -570.60
6471.32 6406.94

AS Trees
N
Mean Residual
SD
Min Residual
Max Residual

633
3.82
54.19
-673.24
318.14

633
-28.17
127.68
-500.68
560.18

633
-106.50
132.23
-676.86
361.19

4.5 Discussion
4.5.1 Density
The dominant stem density model forms were diﬀerent from the random branch path model
forms for all six species sampled in the dataset. Random branch path densities were fairly
82

similar in magnitude compared to dominant stem densities because most of the proﬁle was
shared between the random branch and dominant paths (average RHD was 0.702 for all
trees), except near the top of the relative path length. For four of the six species, the
random branch path density was considerably greater than the dominant stem path near the
top of the relative path length indicating that the branches were denser than the dominant
stem. Okai and Yeboah (2003) also found the density of branch wood to be greater than
stem wood for two tropical hardwood species. However, Swenson and Enquist (2008) found a
high correlation between the dominant stem and branch wood density indicating that branch
wood densities could be used as a proxy for dominant stem wood density.
Wood density has been identiﬁed as an important predictor variable in aboveground
biomass estimation (Chave et al. 2005, Woodall et al. 2011). Chave et al. (2005) used
a species-speciﬁc constant wood density or recommended a stand-level average if speciesspeciﬁc values were unavailable for aboveground biomass estimation across the tropics.
Woodall et al. (2011) also used published species-speciﬁc constant wood density values for
the FIA’s estimation of aboveground biomass. Given the complex pattern of wood density
seen here, the aboveground biomass estimates would be overestimated for the lower half of
a sugar maple tree and underestimated for the upper half. In the case of American beech,
aboveground biomass is underestimated using the species constant value for the entire vertical proﬁle. Branches comprised on average 42 percent of the total volume of study trees,
so that the random branch proﬁle indicated branches to be denser than the dominant stem
would also contribute to greater aboveground biomass estimates. However, the denser random branch proﬁle was near the end of the proﬁle where the volume of branches contributes
small additional amounts to the total volume of a tree.
American beech and sugar maple are both late-successional species in a northern hardwood forest (Dickmann and Leefers 2003). Each of the sampled species came from within
the same stand and experienced the same site conditions, so it was hypothesized that both
species would have a similar vertical density proﬁle shape. However, the vertical density

83

proﬁles for American beech and sugar maple were quite diﬀerent in shape from one another
indicating a species dependence. Repola (2006) also found the density of Scots pine, Norway
spruce, and birch from the same forest type to be species dependent upon vertical location
to varying degrees. The vertical density proﬁles seen here were more complex than a general
increasing or decreasing density pattern in the longitudinal direction.
The magnitude of the dominant stem and random branch path tree density models could
be slightly greater than a wood density model due to the inclusion of bark with the wood.
The bark vertical density models for ﬁve of the six species, except for slippery elm, had much
greater densities than the dominant stem models along the entire vertical proﬁle. However,
bark on average only contributed 6.5 percent to the total green volume of a given sample disc,
without accounting for bark voids. Miles and Smith (2009) report a similar percentage for
bark volume of total volume of 5.7 percent for beech, and a slightly greater percentage of 13.0
percent for general hardwoods. Although the contribution of bark by volume is very small
to the tree density models, the separation of bark from wood is recommended at the onset
of measurements to remove this possible diﬀerence. Both Koch (1970) and Manwiller (1979)
used water saturation of bark samples to determine the green volume before estimating bark
speciﬁc gravity. Koch (1970) found hardwood trees with DBH from 10 to 40 cm, which
is below the mean of the sample trees, to have bark speciﬁc gravity dependent upon the
species, and the pattern with height is also species dependent. The variation in bark speciﬁc
gravity with height within DBH classes was generally greater than the variation at a speciﬁc
height between DBH classes. The speciﬁc gravity increased with height before leveling oﬀ
to the maximum height investigated, unlike the sharp declines that were seen in ﬁve of the
six species in this study. Manwiller (1979) examined smaller diameter (15 cm) hardwoods
on pine sites and found the speciﬁc gravity to be denser than values reported by U. S. D.
A. Forest Service (2010). Signiﬁcant diﬀerences were found among the species for speciﬁc
gravity of bark and among the diﬀerent types of bark (i.e., stem, branch, and tree) for each
species. In this study, the bark densities used to develop the bark proﬁle were a combination

84

of stem bark and branch bark depending upon the selected random path that was followed
for the random branch path proﬁle, so there could be more or less branch bark included in
the bark proﬁle.
An area for further examination would be to extend this study for each species with larger
sample sizes across a broader geographic region. As studies have shown species, geographic
variation, age, and size to be signiﬁcant variables along with the longitudinal gradient of a
tree (Tasissa and Burkhart 1998, Antony et al. 2010). Bayesian statistics and linear model
theory can then be applied to calibrate a broad geographic region’s generalized linear mixedeﬀects model to a speciﬁc forest stand (Lappi 1991). Repola (2006) predicted the random
eﬀects of models by measuring two densities of an individual tree to calibrate models for
pine, spruce, and birch stems in Finland.

4.5.2 Mass
The non-linear mixed-eﬀects models for dominant stem and whole-tree cumulative mass
proﬁles resulted in the smallest or nearly equally small residuals compared to the densityintegral and constant density methods. The non-linear mixed-eﬀects models were compatible
with the total estimates of dominant stem and whole-tree mass by constraining the model,
and the density-integral and constant density models were incompatible with the total mass
estimates. The non-linear mixed-eﬀects proﬁles for dominant stem cumulative mass require
total dominant stem mass and total height as predictor variables into the ﬁnal model, which
total dominant stem mass would require an allometric equation as input into the model with
measurable attributes. The same predictor variables are needed for the whole-tree cumulative
mass proﬁle with the additions of total whole-tree mass, relative crown height, and the mass
at relative crown height. The need for the additional value for mass at relative crown height
as a predictor variable for the whole-tree proﬁle would require measurements of volume with
a dendrometer on a standing tree and the use of a density proﬁle to convert volume to a
mass estimate. Total whole-tree mass would also require the need for an allometric scaling

85

relationship. The use of both the dominant stem and whole-tree proﬁles allows for the mass
of branches to be estimated by simple subtraction rather than a separate model for branches.
Several studies oﬀer solutions to the need for an estimate of total whole-tree mass as
an input into the proﬁle model. Cannell (1984) oﬀers a possible solution for measurable
attributes of diameter, height, form factor, and density as predictor variables for total mass.
Chave et al. (2005) applied a similar model for total aboveground biomass in the tropics
with the same predictor variables except making form factor a constant. MacFarlane and
Ver Planck (2012) also examined the possibility of measuring form factor as an additional
measurement to improve whole-tree mass estimation.
A species-speciﬁc model was shown to give better individual ﬁts for the cumulative dominant stem and whole-tree proﬁles for the American beech and sugar maple trees. The inherent diﬀerences in density among species can be attributed to the need for species-speciﬁc
models for mass estimation. Often single species are the focus of biomass studies in order
to get a large and representative sample size (e.g., Parresol and Thomas 1989, Jordan et
al. 2006). A suggestion for future biomass studies would be to focus on a single species of
interest.
The density-integral method performed well based on the summary statistics of the residuals for the dominant stem. Of the three methods, the density-integral is most likely to be
implemented with the large availability of taper functions to be combined with needed density proﬁles (Jordan et al. 2006). However, the density-integral performed far worse for the
whole-tree cumulative mass proﬁle above relative crown height. Perhaps this is because the
ﬁrst derivative of the cumulative volume in this dataset was not a true representation of a
whole-tree taper function. As when branches were encountered all of the volume was assumed to be located at that relative height. Van Deusen and Roesch (2011) oﬀer another
method of conceptualizing a whole-tree taper proﬁle by compressing all the branches into a
cylinder and measuring diameters of branches at the same heights along the dominant stem.
Another possibility for the whole-tree density-integral not ﬁtting well above relative crown

86

height was the use of the dominant stem density proﬁle. As the crown portion of the tree
is comprised of both branches and dominant stem, some composite density proﬁle may be
needed to better model the observed data.
The method of applying a constant species-speciﬁc density to a cumulative volume proﬁle
performed the worst of the three methods. The mean residuals for both the dominant stem
and whole-tree mass proﬁles were some of the largest. Total whole-tree mass estimates using
the constant density method also resulted in large under- and over-prediction of the observed
total mass of individual trees. The FIA currently uses constant densities to convert volume
estimates to mass estimates showing the need for the adoption of another method for biomass
estimation (Woodall et al. 2011).

87

CHAPTER 5
CONCLUSION

The non-linear mixed-eﬀects models for volume and mass of dominant stems were compatible
to the total volume and mass estimates from Smalian’s formula and random branch sampling.
This is a very important principle for estimates that are generated from models, especially
when the estimates are of economic importance such as whole-tree utilization for energy or
estimating carbon stocks for potential carbon markets. The current study also added to the
limited amount of research focusing on tree branches in relation to a whole tree framework.
A species-composite model was shown to be suﬃcient for a cumulative volume proﬁle.
The composite model was acceptable for volume based on all the trees being sampled from
windfall coming from the same stand and with similar tree architecture, which may have
resulted in these trees mortality. Examination of the sampled trees relative to the residual
stand makes it diﬃcult to apply these models to the residual trees in the stand as it appears
more larger trees were destructively sampled compared to the residual trees in the stand.
It would be interesting to relate the centroid of volume of these wind-thrown trees to the
residual trees, and perhaps because of a higher centroid these trees were more susceptible to
the wind. Newer technologies are making parameters easier to measure on the residual trees
or other standing trees that once were impractical, such as volume at relative crown height.
Species-speciﬁc density models gave the best ﬁt, which makes sense based upon the large
diﬀerences in density between species, e.g., the light wood of American basswood and denser
wood of American beech. Densities of a given tree or stand were shown to be much more
variable than assuming a constant value based on species. For future studies, it is highly
recommended to separate bark from wood at the onset due to major diﬀerences in structure
and density.
The cumulative mass models also had better ﬁts with a species-speciﬁc model. The

88

current research oﬀers a good starting point for whole-tree mass proﬁle modeling with needed
improvements in capturing the various densities of branches. The density-integral approach is
more practical for implementation than the non-linear mixed-eﬀects models, but the densityintegral approach for whole-tree mass was only slightly better than assuming a constant
density approach. A more representative density proﬁle could drastically improve the wholetree mass estimates. An area for further examination would be to sample multiple branch
paths per tree to see how diﬀerent each branch path is from one another and the dominant
stem path. It would aid in determining if a composite density model is needed for the
whole-tree portion of mass modeling above relative crown height.

89

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