Directory UMM :Data Elmu:jurnal:T:Tree Physiology:vol17.1997:
Tree Physiology 17, 71--80
© 1997 Heron Publishing----Victoria, Canada
Crown architecture of Abies balsamea from four canopy positions
DANIEL W. GILMORE1 and ROBERT S. SEYMOUR2
1
Canadian Forest Products Limited, Postal Bag 100, Grande Prairie, Alberta T8V 3A3, Canada
2
Department of Forest Ecosystem Science, College of Natural Resources, Forestry and Agriculture, University of Maine, Orono, ME 04469-5755, USA
Received August 4, 1995
Summary Data collected from four distinct canopy positions
from each of 39 Abies balsamea (L.) Miller trees were used to
construct models to describe the cumulative leaf area distribution within the crown and to predict the needle mass of individual branches, the average branch angle, branch diameter,
branch length, and crown radius per whorl, and the average
number of living branches per whorl. We tested the hypotheses
that regression models are equal among canopy positions and
that a model to predict branch needle mass is valid at the
northern and southern extremes of the central climatic zone of
Maine. Canopy position had an effect on the models constructed to predict needle mass, branch angle, branch diameter,
branch length, crown radius, and the number of living branches
per whorl. However, compared with an expanded model that
incorporated parameters calculated for each crown class, there
was only a small loss in model precision when a general model
constructed from data pooled from all crown classes was used
to predict needle mass, branch angle, and branch diameter.
Regression equations unique to each crown class were needed
to predict crown shape and leaf area distribution in the crown
satisfactorily. Our branch needle mass model, which was constructed from data collected at the southern extreme of the
central climatic zone of Maine, consistently underestimated
needle branch mass when applied to the northern extreme of
the central climatic zone.
Keywords: balsam fir, branch angle, crown shape, foliage
distribution, Maine, needle mass, projected leaf area.
Introduction
Morphological distinctions in the crown architecture of forestgrown trees are used as criteria in formulating silvicultural
prescriptions during various stages of stand development
(Smith 1986). In describing crown architecture, however, it is
not practical to measure every branch on each tree. Thus,
information from models that predict crown radius and needle
mass for individual branches is used in the construction of
other models that describe crown shape, vertical distribution of
leaf area within the crown, and foliage/sapwood area relationships. Numerous model forms have been used to describe
crown attributes. Although additional model forms can be
conceived, it may be more useful to explore the applicability
of model forms that have been documented in previous studies
of crown architecture and modify them as required. The aim of
this study was to use the best model from published studies of
crown architecture to construct regression equations that predict the needle mass of individual branches, the average branch
angle, branch diameter, branch length, and crown radius per
whorl, and the average number of living branches per whorl,
and to describe the cumulative leaf area distribution within the
crown. We also tested the hypothesis that selected equations
were equal among four distinct canopy positions for Abies
balsamea (L.) Miller.
Materials and methods
Study area
Individual trees were selected from the University of Maine
Dwight B. Demeritt Forest and the Penobscot Experimental
Forest, both of which are within a 10-km radius of Orono
(44°54′ N, 68°38′ W) and border the central and southern
climatic zones of Maine (Briggs and Lemin 1992). Soils developed on a parent material of glacial origin (Rourke et al. 1978)
are variable, but are predominantly classified as coarse loamy,
mixed, frigid, Aquic or Typic Haplorthods (Soil Survey Staff
1990), or Gleyed or Orthic Ferro-Humic Podzols (Canadian
Soil Survey Committee 1978). Soil drainage class consistently
bordered between poorly drained and moderately well drained
(Maine Association of Professional Soil Scientists 1990).
Abies--Picea stands were even aged, with A. balsamea being
the primary species. Basal area ranged from 2 to 38 m2 ha −1;
average quadratic diameter at breast height (dbh; measured at
1.3 m above the ground) ranged from 5.7 to 19.0 cm, and
relative density (Curtis 1982) ranged from 4.6 to 11.3 (metric
scale). Sites were rated intermediate in productivity (Briggs
and Lemin 1994), with site index varying from 15 to 20 m at a
breast height (bh) index age of 50 years (Steinman 1992).
Data collection
Stem analysis data were collected from 39 A. balsamea trees
having nearly symmetrical crowns characteristic of four crown
classes (open-grown (OG), codominant (CD), intermediate
(IT) and suppressed (SP), Tables 1 and 2) in July and August
of 1992 and 1993, two or more weeks after bud set. Branch
72
GILMORE AND SEYMOUR
basal diameter (BDIA, cm), average branch length per whorl
(BL, m), branch angle (BANG, degree), and depth in the crown
(L, m) were measured for each branch (Figure 1), and the
number of living branches per whorl (NBW) was counted for
all whorls for each tree. The crown radius (CR, m) was determined, and the base of the live crown (BLC, m) was defined as
the lowest whorl having three living branches. All of the whorl
branches from five open-grown and seven suppressed trees
were sampled. For the remaining trees (four OG, ten CD, ten
IT, three SP), one branch was randomly selected for sampling
from the sixth whorl, as counted from the tree top down
(Figure 1), from the BLC, and from the whorl midway between
the sixth whorl and BLC. All sample branches were oven-dried
at 65 °C for two days. Needles were separated, needles and
branches were re-dried, and the oven-dry branch needle mass
(BNM, g) was determined. The fresh mass for interwhorl
branches was determined in the field and a composite sample
of interwhorl branches collected from throughout the crown
was oven-dried to obtain an oven-dry/fresh interwhorl mass
ratio per tree. A needle/total branch mass ratio was calculated
from the whorl branches sampled for each tree and used to
estimate the needle mass of the interwhorl branches from their
fresh mass. The average percentage (range in parentheses) of
oven-dry foliage mass in interwhorl branches for each crown
class was: OG 33% (12--67%); CD 33% (17--71%); IT 16%
(0--24%); and SP 13% (0--35%). In an earlier study (Gilmore
et al. 1995), we found that between-tree variation in specific
leaf area (SLA, projected fresh leaf area per kg oven-dry
foliage) was greater than within-tree variation in SLA, and
demonstrated that the use of an average SLA per tree to
Figure 1. Schematic diagram of branch length (BLi) and crown radius
(CRi) predictor variables. H = tree height, Li = branch basal distance
from tree top, VLi = CRi distance to tree top.
Table 1. Morphological characteristics of sample trees from each canopy position. Ranges of data are shown in parentheses.
Crown class
n
dbh1 (cm)
Height (m)
Total age (years)
bh age1 (years)
Live crown ratio (%)
Open-grown
Codominant
Intermediate
Suppressed
9
10
10
10
15.2 (5.7--30.5)
20.2 (9.5--29.7)
10.8 (5.3--14.5)
6.2 (2.5--10.2)
9.3 (4.4--16.6)
16.6 (7.6--19.6)
12.7 (7.2--16.0)
7.6 (4.0--12.8)
53 (20--81)
592 (25--84)
502 (22--79)
422 (17--63)
27 (11--60)
47 (18--65)
42 (18--62)
33 (10--47)
85 (76--89)
47 (31--71)
29 (16--43)
21 (9--43)
1
2
Measured at 1.3 m.
Total age data not available for one tree because of decayed heartwood.
Table 2. Summary statistics for average branch attributes by crown class.
Variable1
Crown class
Open-grown
BDIA (cm)
BNM (g)
BANG (degrees)
BL (m)
CR (m)
1
Codominant
Intermediate
Suppressed
n
Mean
SE
n
Mean
SE
n
Mean
SE
n
Mean
SE
80
80
628
628
620
1.74
81.37
64.90
1.45
1.30
0.070
10.871
0.780
0.030
0.030
31
31
799
797
796
1.82
100.68
58.60
1.13
0.99
0.135
17.306
0.594
0.024
0.024
18
18
500
500
501
0.98
45.21
56.10
0.70
0.59
0.064
6.032
0.693
0.016
0.016
274
274
349
349
349
0.62
14.73
51.30
0.63
0.60
0.021
1.391
1.095
0.025
0.025
Variables are: BDIA = branch basal diameter; BNM = branch needle mass; BANG = branch angle; BL = average branch length per whorl;
CR = crown radius.
CROWN ARCHITECTURE OF ABIES BALSAMEA
determine the total projected leaf area of a tree was statistically
valid. Therefore, an average SLA for each tree was calculated
from two 10-needle samples collected from each of five foliage
age classes (current-year, 1-year-old, 2-year-old, 3-year-old
and 4-year-old-plus) for each randomly selected branch per
tree.
To test the applicability of the BNM prediction model at the
northern extreme of the central climatic zone of Maine, data
were collected from 36 branches from 12 codominant A. balsamea trees fom a 23-year-old fir--spruce stand in Bald Mountain Township, Maine during the third week of August 1993.
The study site was located approximately 100 km northwest of
Orono in the northern portion of the central climatic zone of
Maine (Briggs and Lemin 1992) and is fully described elsewhere (Newton et al. 1989, 1992).
Model formulation
Several published model forms were screened in their original
and mathematically modified forms to predict BNM, BANG,
NBW, BDIA, BL, and crown radius (CR). Models to predict
BL and CR were fit to the average measurements per whorl
collected from the tree top down to the widest portion of the
crown. Needle mass was predicted from BDIA for all whorls,
and the needle mass for all interwhorl branches was estimated
from the oven-dry/fresh interwhorl mass ratio for each tree.
Mean branch diameters (cm, ± SE) used to predict needle mass
for trees of each crown class were: OG 1.91 ± 0.033, CD 1.69
± 0.024, IT 0.97 ± 0.015, and SP 0.63 ± 0.022. Total projected
leaf area (TOTLA, m2) for each tree was calculated by multiplying the average SLA for each tree by the total estimated
needle mass per tree.
A cumulative distribution function was fit by nonlinear least
squares regression to describe the relative leaf area distribution
within the crown. For each tree, projected leaf area (PLA, m2)
was summed for each of 10 equally spaced crown sections
from the tree top down to the lowest living branch. According
to our definition of live crown base, it was possible to have
living branches below the BLC. The relative leaf area for each
crown section (LAi/TOTLA, where i = crown section) was
obtained by dividing the leaf area per crown section by TOTLA
per tree; therefore, the cumulative leaf area distribution for
each tree was equal to one.
Model selection
Model selections were based on scatterplots of residuals versus
predicted values in combination with a modified likelihood
criterion that was equal to the root mean squared error from
unweighted and untransformed equations, and to a rescaled
root mean squared error from weighted and transformed equations (Furnival 1961). Coefficients of determination (R2) were
not used as selection criteria when transformations were performed on the dependent variables. Because an adjustment for
logarithmic bias should be considered when applying results
from logarithmically transformed models (Finney 1941,
Snowdon 1991), the ratio estimator correction factor for logarithmic bias (Snowdon 1991) was applied to all estimates used
73
in subsequent analyses and in calculations of bias (bias =
(observed − predicted)/n) in the original units of measure.
Parameter estimation
Analysis of covariance (ANOCOVA, Zar 1984) was used to
test the hypothesis that a general model applicable to each
crown class would adequately describe a relationship. When
warranted, regression with indicator variables (Neter et al.
1990) was used to construct an expanded equation applicable
to each crown class. The completely expanded model can be
thought of as a combination of four equations with the parameter estimates for the CD, IT, and SP crown classes depicting
their respective differences from the OG class. The results of
these analyses can be defined by two extremes. If canopy
position had no effect on a model, a single equation would
adequately describe the relationship. Conversely, a separate
equation would be required for each crown class if pronounced
differences for a given relationship existed among canopy
positions. In this paper, only the parameter estimates in their
reduced form are presented for the models having the best fit
to describe a particular relationship. Parameter estimates, fit
statistics for the completely expanded and intermediate models, and an explicit statement of the hypotheses tested during
the construction of the final models are provided in Gilmore
(1995). All statistical analyses were done using SYSTAT
(SYSTAT, Inc. 1992, Evanston, IL).
Results
The selected models, their parameter estimates, and their respective indices of fit are presented in Table 3. The models
having best indices of fit, and thus best satisfying the assumption of least squares regression while accounting for differences in scale (Furnival 1961) are marked with an asterisk.
Branch needle mass prediction
The selected BNM prediction model was a logarithmic transformation of a model presented by Ek (1979) (Table 3). We
rejected the hypothesis (P < 0.001) that a common BNM
prediction model could be used for all crown classes. However,
comparison of the fit statistics for the general and expanded
versions of this model revealed only a small sacrifice in overall
precision for the general model compared with the expanded
model (Table 4). Therefore, because of its simplicity, we recommend Model 1 (Table 5) over the expanded model. Relative to the upper canopy positions (OG and CD), bias for the
modeling data was greatest for the prediction of needle mass
of branches from IT trees and small branches from SP trees
(Table 5, Figure 2a).
We tested the applicability of the BNM prediction model
(Model 1) constructed from data at the southern extreme of the
central climatic zone in Maine with independent data collected
at the northern extreme of this climatic zone. The BNM prediction model consistently underestimated BNM (Figure 3)
and, when we corrected for logarithmic bias, predicted values
had an average bias of 40%.
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GILMORE AND SEYMOUR
Table 3. Models screened to predict branch needle mass, number of branches per whorl, and branch diameter. Standard errors and probability values
of parameter estimates are shown in parentheses; selected models are marked with an asterisk.
Model form1
MSE
R2
Furnival’s index
BNM = b0 + b1BDIA2
b0 = 5.979 (SE = 2.821, P = 0.034)
b1 = 21.896 (SE = 1.183, P < 0.001)
2153.091
0.459
46.40
0.798
0.748
10.22
2103.381
0.473
45.86
0.729
0.767
9.77
1960.753
0.510
44.28
1.153
0.635
12.28
2853.649
0.289
53.42
4
NBW = ln(b0) + b1ln(GI)
b0 = 4.310 (SE = 0.086, P < 0.001)
b1 = 0.834 (SE = 0.049, P < 0.001)
1.203
0.292
1.09
*5NBW = b0GI b1 [exp(b2L)]
b0 = 4.979 (SE = 0.199, P < 0.001)
b1 = 0.319 (SE = 0.021, P < 0.001)
b2 = −0.017 (SE = 0.006, P < 0.001)
1.168
0.314
1.05
Average BDIA = b1L + b2L2
b1 = 0.531 (SE = 0.011, P < 0.001)
b2 = −0.025 (SE = 0.001, P < 0.001)
0.200
0.913
0.44
*Average BDIA = b1dbh + b2L + b3L2
b1 = 0.039 (SE = 0.001, P < 0.001)
b2 = 0.301 (SE = 0.012, P < 0.001)
b3 = − 0.011 (SE = 0.001, P < 0.001)
0.104
0.955
0.32
*ln(BNM) = b0 + b1ln(BDIA)
b0 = 3.068 (SE = 0.048, P < 0.001)
b1 = 2.134 (SE = 0.062, P < 0.001)
2
BNM = b0BDIA b1
b0 = 32.443 (SE = 2.736, P < 0.001)
b1 = 1.609 (SE = 0.097, P < 0.001)
ln(BNM) = b0 + b1ln(BDIA) + b2ln(L)
b0 = 2.841 (SE = 0.059, P < 0.001)
b1 = 1.545 (SE = 0.110, P < 0.001)
b2 = 0.415 (SE = 0.066, P < 0.001)
2
BNM = b0BDIA b1Lb2
b0 = 20.773 (SE = 2.840, P < 0.001)
b1 = 1.237 (SE = 0.129, P < 0.001)
b2 = 0.487 (SE = 0.091, P < 0.001)
3
ln(BNM) = b0 + b1ln(BDIA × WA) + b2ln(NW)
b0 = 1.981 (SE = 0.467, P < 0.001)
b1 = 1.060 (SE = 0.040, P < 0.001)
b2 = −0.508 (SE = 0.160, P = 0.001)
BNM = b0(BDIA × WA)b1NW b2
b0 = 26.914 (SE = 13.280, P < 0.001)
b1 = 1.003 (SE = 0.102, P < 0.001)
b2 = −0.748 (SE = 0.202, P < 0.001)
1
2
3
4
5
BNM = branch needle mass (g), BDIA = branch basal diameter (cm), L = distance from branch base to tree top (m), H = total tree height (m),
dbh = diameter at breast height (cm, measured at 1.3 m), WA = whorl age, NW = total number of whorls in the crown.
From Ek (1979).
From Kleinschmidt et al. (1980).
From Powell (1977).
Adapted from Maguire et al. (1994).
Branch angle prediction
We explored the use of the model presented by Colin and
Houllier (1992) (Table 5), to predict BANG. Although we
rejected the hypothesis that a general model adequately de-
scribed the BANG relationship for each crown class, we present only a general model (Model 2) because, as a result of the
variation in our data and the mathematical complexity of the
model, we were not able to fit a model with indicator variables
to obtain parameters for each crown class. Predicted values for
CROWN ARCHITECTURE OF ABIES BALSAMEA
75
Table 4. Comparison of fit statistics for general prediction models applied to the pooled data for each crown class and expanded models where
parameters were calculated for each crown class. Abbreviations: BNM = branch needle mass; BANG = branch angle; BDIA = branch basal
diameter; BL = average branch length per whorl; CR = crown radius; LA/TOTLA = crown section relative leaf area; GI = previous year height
increment; dbh = diameter at breast height; L = depth in to the crown; VL as defined in Figure 1; and H = tree height.
Model
Dependent
variable
Predictor
variables
General
R2
Expanded
R2
General
MSE
Expanded
MSE
1
2
2
3
4
4
53
53
53
64
64
7
BNM
BANG average
BANG maximum
NBW
BDIA average
BDIA maximum
BL
CR average
CR maximum
CR average
CR maximum
LA/TOTLA
BDIA
Whorl age
Whorl age
GI, L
dbh, L
dbh, L
L
VL
VL
L, H
L, H
VL
0.7481
0.419
0.437
0.314
0.8161
0.7971
0.831
0.811
0.826
0.789
0.795
0.884
0.754
na2
na2
na2
0.819
0.798
0.8871
0.8601
0.8661
0.8751
0.8731
0.9111
0.7981
179.647
182.422
1.168
0.1031
0.1451
0.068
0.072
0.091
0.069
0.091
0.015
0.799
na2
na2
na2
0.102
0.144
0.0461
0.0541
0.0641
0.0491
0.0611
0.0121
1
2
3
4
Denotes the form (general or expanded) of the model selected for presentation.
An expanded model was not fit with indicator variables for reasons discussed in text.
From Mitchell (1975) model form.
From Honer (1971) model form.
Table 5. Parameter estimates and bias (expressed as a percentage of the mean for each crown class in parenthesis) among crown classes for final
models to predict branch needle mass (BNM), branch angle (BANG), the number of branches per whorl (NBW), and branch diameter (BDIA) for
Abies balsamea. Other abbreviations: WA = whorl age; and L = depth into the crown.
Model
11
Final model form
Bias
ln(BNM) = 3.068 + 2.134[ln(BDIA)]
MSE = 0.798, R2 = 0.748
Open-grown
Codominant
Intermediate
Suppressed
− 11.0 (−13.5%)
− 5.1 (−5.0%)
19.4 (42.9%)
2.5 (16.9%)
2
Average BANG = 39.296 + 51.609[1 − exp(−0.059WA)]
MSE = 179.647, R2 = 0.419
− 0.6 (−0.9%)
− 2.1 (−3.5%)
1.9 (3.0%)
1.5 (2.2%)
2
Max BANG = 43.049 + 52.404[1 − exp(− 0.064WA)]
MSE = 182.422, R2 = 0.437
− 0.6 (−0.9%)
− 1.8 (−2.7%)
1.3 (1.9%)
1.7 (2.3%)
3
NBW = 4.979GI
[exp(−0.017L)]
MSE = 1.168, R2 = 0.314
0.3 (8.3%)
0.1 (2.8%)
4
Average BDIA = 0.039dbh + 0.301L − 0.011L
MSE = 0.104, R2 = 0.816
0.2 (0.3%)
− 0.1 (−0.1%)
− 0.1 (−0.1%)
0.1 (0.1%)
4
Max BDIA = 0.041dbh + 0.349L − 0.014L
MSE = 0.145, R2 = 0.717
0.3 (14.1%)
− 0.1 (−9.0%)
− 0.1 (−5.5%)
0.1 (15.1%)
1
0.319
0 (0%)
−0.4 (− 21.5%)
2
2
Ratio correction factor for logarithmic bias = 1.144.
BANG increased as branches aged and their mass increased
with the accumulation of branchwood and needle mass (Table 5). Whorl age (WA, years) was the variable with the highest
correlation with BANG (r = 0.62, P < 0.001 and r = 0.63,
P < 0.001 with average and maximum BANG, respectively).
Other variables, such as dbh, depth into the crown (L), crown
length (CL, m), tree height (H), whorl height and branch length
(BL) had weak correlations (r < 0.48) with average or maximum BANG, and when included in various forms of the
model, their parameter estimates did not differ from zero.
Number of branches per whorl prediction
A variation of the model originally presented by Maguire et al.
(1994) to predict NBW had the best index of fit (Table 3).
Because the precision with which the model (Model 3, Table 5) predicted NBW was low (R2 = 0.314), we did not attempt
to obtain parameter estimates for each crown class with indicator variables. The absolute bias for the general NBW prediction model for each crown class was also low, however, and
bias expressed as a percentage of the mean was less than 10%
for all but the SP crown class (Table 5). Additional condition-
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GILMORE AND SEYMOUR
Figure 3. Plotted regression line for Model 1 constructed from data
from the southern extreme of the central climatic zone of Maine and
the natural logarithm of observed branch needle mass values from
Bald Mountain Township located at the northern extreme of the central
climatic zone.
Figure 2. (A) Log-log plot of needle mass (BNM, g) versus branch
diameter (BDIA, cm) and the predictive function ln(BNM) = 3.068 +
2.134 ln(BDIA). (B) Estimates of ln(BNM) from Model 3 using
ln(BDIA) as the sole predictor versus results from the model:
ln(BNM) = 2.841 + 1.545 ln(BDIA) + 0.415 ln(L).
ing of the NBW prediction model with predictor variables to
represent tree size parameters such as dbh, H, and CL did not
improve precision.
Branch diameter prediction
We rejected the hypothesis (P < 0.001) that a common BDIA
prediction model could be used for all crown classes. Because
differences in fit statistics between the general and expanded
models were minor (Table 4), we present the general model
constructed from data pooled from all four canopy positions.
The equations constructed to predict average and maximum
BDIA from L and dbh had low bias among crown classes
(Table 5).
Branch length and crown radius prediction
We rejected the hypotheses (P < 0.001) that a common BL or
CR prediction model could be used for all crown classes. The
expanded forms of the models of Mitchell ((1975), Model 5)
and Honer ((1971), Model 6) to predict CR produced similar
R2 values and MSEs (Table 4). A comparison of the models
using the maximum CL (8 m) for a hypothetical open-grown
tree (Figure 4) indicated that crown radius estimates obtained
from Honer’s and Mitchell’s models were identical, but the
bias of Honer’s model was less (Table 6). Each of these model
forms predicted crown radii within 0.5 m of those predicted
with Honer’s (1971) original coefficients (Figure 4). Mitchell’s
model ((1975), Model 5) is simpler, requiring only one independent variable, but CR will always be predicted to increase
with increasing L because of the logarithmic transformation
incorporated into the model (Figure 5a). Honer’s model
((1971), Model 6) is more complex, requiring two independent
variables and the calculation of several interaction terms. Both
model forms behave rationally when applied to values within
the range of data from which they were constructed (Table 2,
Figures 4 and 5).
Leaf area distribution
We rejected the hypothesis (P < 0.001) that a common cumulative leaf area distribution model could be used for all crown
classes (Table 6, Figure 6a). The cumulative leaf area distribution for each tree was always one, but because there was
considerable variation in the cumulative leaf area distribution
at relative depths into the crown among trees, Model 7 never
predicted a value of one for cumulative leaf area. To visualize
the predicted relative leaf area distribution for each crown
class, Figure 6b was constructed by predicting the cumulative
leaf distribution with Model 7 (Table 6) and subtracting the
predicted leaf area distribution in the upper crown sections
from each progressively lower crown section. The leaf area
distribution in the crown at the lowest living branch, as illustrated by Figure 6b, could not be zero.
Discussion
Several studies have suggested that the inclusion of depth in
the crown (L) in a BNM prediction model improves precision
(Ek 1979, Gillespie et al. 1994). We found that BDIA and L
were highly correlated (r = 0.81, P < 0.001), and therefore, to
avoid multicollinearity, L was not included in Model 1. A
scatter plot of predicted values of ln(BNM) from Model 1
CROWN ARCHITECTURE OF ABIES BALSAMEA
Figure 4. Comparison of the fit of Honer’s (1971) and Mitchell’s
(1975) models to Honer’s (1971) published equation for Abies balsamea for a hypothetical open-grown tree of average crown length.
versus predicted values of ln(BNM) from a model incorporating ln(BDIA) and ln(L) as predictor variables revealed that
ln(L) has a minimal effect on predicted values. The greatest
77
divergence in predicted values between the two models was for
small branches from the SP crown class (Figure 2b).
Many early branch-level biomass prediction models have
not been tested with data collected outside of the geographic
location from which they were constructed (Attiwill 1962,
Loomis et al. 1966, Ek 1979). For A. balsamea, no differences
in parameter estimates among data collection points were
detected between New Brunswick, Canada and New Hampshire, USA during the construction of a branch-level biomass
prediction model (Kleinschmidt et al. 1980). However, differences in parameter estimates among research plantations were
detected for branch-level biomass prediction models constructed for Pseudotsuga menziesii (Mirb.) Franco (Kurz
1989), suggesting that branch-level biomass prediction models
should be verified with independent data collected from outside their region of construction. Our inability to validate
Model 1 across a narrow latitudinal gradient contrasts with the
results from a study of A. balsamea by Kleinschmidt et al.
(1980) who concluded that their BNM prediction model was
applicable in New Hampshire, USA and New Brunswick,
Canada. However, when applied to our data, their model form
had a poor fit relative to our BNM prediction model (Table 3).
Kleinschmidt et al. (1980) did not sample trees from the
Table 6. Parameter estimates and bias (expressed as a percentage of the mean for each crown class in parenthesis) by crown class for the final
models to predict branch length (BL) and crown radius (CR), and to describe the cumulative foliage distribution in Abies balsamea. Variable VL
is as defined in Figure 1. Other variables: H = tree height; and LAi /TOTLA = relative leaf area for each crown section.
Model
Crown class
Final model
51
Open-grown
Codominant
Intermediate
Suppressed
BL = 1.783ln[(L/3.248) + 1]
BL = 1.783ln[(L/4.023) + 1]
BL = 0.736ln[(L/1.260) + 1]
BL = 1.783ln[(L/2.258) + 1]
0.006
0.001
0.001
0.014
(0.4%)
(0.1%)
(0.1%)
(2.2%)
51
Open-grown
Codominant
Intermediate
Suppressed
Average CR = 1.763ln[(VL/3.271) + 1]
Average CR = 1.763ln[(VL/4.015) + 1]
Average CR = 0.728ln[(VL/1.193) + 1]
Average CR = 2.349ln[(VL/3.271) + 1]
0.438
0.323
0.244
0.139
(33.6%)
(32.6%)
(41.3%)
(23.1%)
51
Open-grown
Codominant
Intermediate
Suppressed
Max CR = 2.048ln[(VL/3.444) + 1]
Max CR = 2.048ln[(VL/4.187) + 1]
Max CR = 1.098ln[(VL/1.758) + 1]
Max CR = 1.098ln[(VL/0.997) + 1]
0.013
0.002
−0.123
0.124
(9.6%)
(0.2%)
(− 18.1%)
(18.3%)
62
Open-grown
Codominant
Intermediate
Suppressed
Average CR = 0.761VL − 0.025VLH − 0.604VL2/H − 0.028VL2
Average CR = 0.490VL − 0.007VLH − 0.220VL2/H − 0.003VL2
Average CR = 0.761VL − 0.025VLH − 1.665VL2/H + 0.087VL2
Average CR = 0.761VL − 0.009VLH + 0.830VL2/H − 0.182VL2
0.027
0.021
0.042
0.062
(2.1%)
(2.1%)
(7.1%)
(10.3%)
62
Open-grown
Codominant
Intermediate
Suppressed
Max CR = 0.847VL − 0.030VLH − 0.668VL2/H + 0.034VL2
Max CR = 0.391VL + 0.001VLH − 0.003VL2/H − 0.015VL2
Max CR = 0.604VL − 0.006VLH − 0.668VL2/H + 0.006VL2
Max CR = 0.847VL − 0.004VLH + 0.967VL2/H − 0.226VL2
0.019
0.019
0.017
0.047
(1.4%)
(1.8%)
(2.5%)
(6.9%)
7
Open-grown
Codominant
Intermediate
Suppressed
LAi /TOTLA = 1 − exp(− 2.748VL3.027 )
LAi /TOTLA = 1 − exp(− 3.610VL3.027 )
LAi /TOTLA = 1 − exp(− 2.748VL1.885 )
LAi /TOTLA = 1 − exp(− 2.748VL2.231 )
0.008
0.005
0.008
0.005
(0.8%)
(0.5%)
(0.8%)
(0.5%)
1
2
Model 5 = Mitchell (1975) model form.
Model 6 = Honer (1971) model form.
Bias
78
GILMORE AND SEYMOUR
Figure 5. Relative comparison of (A) Mitchell (1975) and (B) Honer
(1971) model forms among crown classes for hypothetical trees of
average crown length.
Figure 6. Predicted (A) cumulative and (B) relative leaf area distribution within the crown of Abies balsamea by crown class.
intermediate or suppressed crown classes. Kershaw (1993)
found differences in a BNM model constructed for Tsuga
heterophylla (Raf.) Sarg. among research plantations established in central Washington, USA and speculated that these
differences were attributable to differences in needle mass of
trees between regions. Herbicide trials (Newton et al. 1989,
1992) may have reduced competition enough to influence
branch density (number of branches per unit crown length) in
the trees used to test the applicability of our model to a
different region. Additionally, site quality may have influenced
the performance of the BNM prediction model outside of the
study region. Potential site productivity for A. balsamea in
Bald Mountain Township, Maine is less than that of our study
area (R.D. Briggs, personal communication). Because of the
poor performance of the BNM prediction model outside our
study area, we conclude that branch-level biomass prediction
models should be verified before widespread application.
Colin and Houllier (1992) had difficulties predicting BANG
for plantation-grown Picea abies Karst., and suggested that
unmeasured factors, such as site quality and genetics, may
influence branch angle. Although we took care to sample trees
from sites of comparable quality, sampling from a naturally
regenerated population precludes control over genetic variation among trees.
Maguire et al. (1994) included relative stand dbh as a predictor variable in their original version of Model 3 to predict
NBW to account for social position. Crown class influenced
Model 3, but parameter estimates for other variables (dbh, H)
included in the model to represent tree size did not differ from
zero. Previous-year height increments (GI, m) characteristic to
each crown class may have contributed to our inability to
expand Model 3 mathematically.
Large diameter trees were predicted to have greater BDIAs
relative to small diameter trees. Inspection of the quadratic
term for each Model 4 indicates that BDIA will eventually
decrease with depth into the crown. This is consistent with
results reported by Colin and Houllier (1992) and Maguire et
al. (1994) for plantation-grown Picea abies and Pseudotsuga
menziesii, respectively. Branch basal diameter (BDIA) growth
in the lower portion of the crowns of forest-grown trees would
be expected to slow as a result of increased mutual shading and
as branches lose their vigor and the non-foliated lower portion
of the crown becomes larger (Mitchell 1975, Jack and Long
1992). At the same time, branches in the upper portion of the
crown would be expected to retain their vigor, and their accumulated BDIA growth would eventually be greater than that in
the lower portions of the crown.
Leaf area of the understory crown classes was concentrated
in the upper 45% of the crown (Figure 6b). Morphologically,
this would increase the ability of understory trees (IT and SP)
to produce photosynthate at the low irradiances that characterize their canopy position. Crowns of SP trees were also
CROWN ARCHITECTURE OF ABIES BALSAMEA
found to be flat relative to other crown classes, with only small
tufts of foliage on their lower branches (Figure 5) and most of
their leaf area concentrated in the upper portion of the crown
(Figure 6). In contrast, leaf area of trees in the OG and CD
crown classes was concentrated in the lower 60% of the crown.
By definition (Smith 1986), OG and CD trees have adequate
growing space; therefore, one would expect older branches in
the lower portion of the crown to have greater leaf area than
younger branches higher in the crown. Crown shape morphology for each canopy position conformed to the Massart model
of crown architecture (Hallé et al. 1978). Kohyama (1980) also
observed patterns of crown architecture in OG and SP Abies
mariesii Mast. saplings that conform to the Massart model.
Measures of crown geometry can be incorporated into sophisticated growth and yield prediction models, such as The
Tree and Stand Simulator (TASS, Mitchell 1975), that use
crown size as a measure of competition, into ecosystem-based
models, such as those derived from JABOWA (Botkin et al.
1972), and into nutrient cycling models (Kimmins 1988) that
use a stratified leaf area distribution in the crown to model
growth efficiency.
Acknowledgments
Financial support for this project was provided by a grant from the
Maine Agricultural and Forest Experiment Station to R.S. Seymour
and by the University of Maine Cooperative Forestry Research Unit.
Appreciation is extended to Michael P. Auger, Kelley A. Brackley, Neil
A. Brackley, Douglas S. Campbell, Anne L. Coyle, Michael E. Day,
Christina Epperson, James Frohn, Stuart W. Gardner, Jason D. Moore,
James T. Murray, Stephen W. Murray, Francine Natale, Andrew L.
Rotch, Perry S. Sawyer, II, Brian D. Tift, David G. Ray, Dale R.
Whitlock, and Kurt E. Zschau for field and laboratory assistance. Data
from Bald Mountain Township were courtesy of Russell D. Briggs,
Richard J. Dionne, Ronald C. Lemin, Jr., and Maxwell L. McCormack, Jr. This manuscript benefited from reviews by Russell D.
Briggs, Marie R. Coyea, Michael S. Greenwood, William A. Halteman, Douglas A. Maguire, and Alan S. White. Publication 2003 of the
Maine Agricultural and Forest Experiment Station.
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© 1997 Heron Publishing----Victoria, Canada
Crown architecture of Abies balsamea from four canopy positions
DANIEL W. GILMORE1 and ROBERT S. SEYMOUR2
1
Canadian Forest Products Limited, Postal Bag 100, Grande Prairie, Alberta T8V 3A3, Canada
2
Department of Forest Ecosystem Science, College of Natural Resources, Forestry and Agriculture, University of Maine, Orono, ME 04469-5755, USA
Received August 4, 1995
Summary Data collected from four distinct canopy positions
from each of 39 Abies balsamea (L.) Miller trees were used to
construct models to describe the cumulative leaf area distribution within the crown and to predict the needle mass of individual branches, the average branch angle, branch diameter,
branch length, and crown radius per whorl, and the average
number of living branches per whorl. We tested the hypotheses
that regression models are equal among canopy positions and
that a model to predict branch needle mass is valid at the
northern and southern extremes of the central climatic zone of
Maine. Canopy position had an effect on the models constructed to predict needle mass, branch angle, branch diameter,
branch length, crown radius, and the number of living branches
per whorl. However, compared with an expanded model that
incorporated parameters calculated for each crown class, there
was only a small loss in model precision when a general model
constructed from data pooled from all crown classes was used
to predict needle mass, branch angle, and branch diameter.
Regression equations unique to each crown class were needed
to predict crown shape and leaf area distribution in the crown
satisfactorily. Our branch needle mass model, which was constructed from data collected at the southern extreme of the
central climatic zone of Maine, consistently underestimated
needle branch mass when applied to the northern extreme of
the central climatic zone.
Keywords: balsam fir, branch angle, crown shape, foliage
distribution, Maine, needle mass, projected leaf area.
Introduction
Morphological distinctions in the crown architecture of forestgrown trees are used as criteria in formulating silvicultural
prescriptions during various stages of stand development
(Smith 1986). In describing crown architecture, however, it is
not practical to measure every branch on each tree. Thus,
information from models that predict crown radius and needle
mass for individual branches is used in the construction of
other models that describe crown shape, vertical distribution of
leaf area within the crown, and foliage/sapwood area relationships. Numerous model forms have been used to describe
crown attributes. Although additional model forms can be
conceived, it may be more useful to explore the applicability
of model forms that have been documented in previous studies
of crown architecture and modify them as required. The aim of
this study was to use the best model from published studies of
crown architecture to construct regression equations that predict the needle mass of individual branches, the average branch
angle, branch diameter, branch length, and crown radius per
whorl, and the average number of living branches per whorl,
and to describe the cumulative leaf area distribution within the
crown. We also tested the hypothesis that selected equations
were equal among four distinct canopy positions for Abies
balsamea (L.) Miller.
Materials and methods
Study area
Individual trees were selected from the University of Maine
Dwight B. Demeritt Forest and the Penobscot Experimental
Forest, both of which are within a 10-km radius of Orono
(44°54′ N, 68°38′ W) and border the central and southern
climatic zones of Maine (Briggs and Lemin 1992). Soils developed on a parent material of glacial origin (Rourke et al. 1978)
are variable, but are predominantly classified as coarse loamy,
mixed, frigid, Aquic or Typic Haplorthods (Soil Survey Staff
1990), or Gleyed or Orthic Ferro-Humic Podzols (Canadian
Soil Survey Committee 1978). Soil drainage class consistently
bordered between poorly drained and moderately well drained
(Maine Association of Professional Soil Scientists 1990).
Abies--Picea stands were even aged, with A. balsamea being
the primary species. Basal area ranged from 2 to 38 m2 ha −1;
average quadratic diameter at breast height (dbh; measured at
1.3 m above the ground) ranged from 5.7 to 19.0 cm, and
relative density (Curtis 1982) ranged from 4.6 to 11.3 (metric
scale). Sites were rated intermediate in productivity (Briggs
and Lemin 1994), with site index varying from 15 to 20 m at a
breast height (bh) index age of 50 years (Steinman 1992).
Data collection
Stem analysis data were collected from 39 A. balsamea trees
having nearly symmetrical crowns characteristic of four crown
classes (open-grown (OG), codominant (CD), intermediate
(IT) and suppressed (SP), Tables 1 and 2) in July and August
of 1992 and 1993, two or more weeks after bud set. Branch
72
GILMORE AND SEYMOUR
basal diameter (BDIA, cm), average branch length per whorl
(BL, m), branch angle (BANG, degree), and depth in the crown
(L, m) were measured for each branch (Figure 1), and the
number of living branches per whorl (NBW) was counted for
all whorls for each tree. The crown radius (CR, m) was determined, and the base of the live crown (BLC, m) was defined as
the lowest whorl having three living branches. All of the whorl
branches from five open-grown and seven suppressed trees
were sampled. For the remaining trees (four OG, ten CD, ten
IT, three SP), one branch was randomly selected for sampling
from the sixth whorl, as counted from the tree top down
(Figure 1), from the BLC, and from the whorl midway between
the sixth whorl and BLC. All sample branches were oven-dried
at 65 °C for two days. Needles were separated, needles and
branches were re-dried, and the oven-dry branch needle mass
(BNM, g) was determined. The fresh mass for interwhorl
branches was determined in the field and a composite sample
of interwhorl branches collected from throughout the crown
was oven-dried to obtain an oven-dry/fresh interwhorl mass
ratio per tree. A needle/total branch mass ratio was calculated
from the whorl branches sampled for each tree and used to
estimate the needle mass of the interwhorl branches from their
fresh mass. The average percentage (range in parentheses) of
oven-dry foliage mass in interwhorl branches for each crown
class was: OG 33% (12--67%); CD 33% (17--71%); IT 16%
(0--24%); and SP 13% (0--35%). In an earlier study (Gilmore
et al. 1995), we found that between-tree variation in specific
leaf area (SLA, projected fresh leaf area per kg oven-dry
foliage) was greater than within-tree variation in SLA, and
demonstrated that the use of an average SLA per tree to
Figure 1. Schematic diagram of branch length (BLi) and crown radius
(CRi) predictor variables. H = tree height, Li = branch basal distance
from tree top, VLi = CRi distance to tree top.
Table 1. Morphological characteristics of sample trees from each canopy position. Ranges of data are shown in parentheses.
Crown class
n
dbh1 (cm)
Height (m)
Total age (years)
bh age1 (years)
Live crown ratio (%)
Open-grown
Codominant
Intermediate
Suppressed
9
10
10
10
15.2 (5.7--30.5)
20.2 (9.5--29.7)
10.8 (5.3--14.5)
6.2 (2.5--10.2)
9.3 (4.4--16.6)
16.6 (7.6--19.6)
12.7 (7.2--16.0)
7.6 (4.0--12.8)
53 (20--81)
592 (25--84)
502 (22--79)
422 (17--63)
27 (11--60)
47 (18--65)
42 (18--62)
33 (10--47)
85 (76--89)
47 (31--71)
29 (16--43)
21 (9--43)
1
2
Measured at 1.3 m.
Total age data not available for one tree because of decayed heartwood.
Table 2. Summary statistics for average branch attributes by crown class.
Variable1
Crown class
Open-grown
BDIA (cm)
BNM (g)
BANG (degrees)
BL (m)
CR (m)
1
Codominant
Intermediate
Suppressed
n
Mean
SE
n
Mean
SE
n
Mean
SE
n
Mean
SE
80
80
628
628
620
1.74
81.37
64.90
1.45
1.30
0.070
10.871
0.780
0.030
0.030
31
31
799
797
796
1.82
100.68
58.60
1.13
0.99
0.135
17.306
0.594
0.024
0.024
18
18
500
500
501
0.98
45.21
56.10
0.70
0.59
0.064
6.032
0.693
0.016
0.016
274
274
349
349
349
0.62
14.73
51.30
0.63
0.60
0.021
1.391
1.095
0.025
0.025
Variables are: BDIA = branch basal diameter; BNM = branch needle mass; BANG = branch angle; BL = average branch length per whorl;
CR = crown radius.
CROWN ARCHITECTURE OF ABIES BALSAMEA
determine the total projected leaf area of a tree was statistically
valid. Therefore, an average SLA for each tree was calculated
from two 10-needle samples collected from each of five foliage
age classes (current-year, 1-year-old, 2-year-old, 3-year-old
and 4-year-old-plus) for each randomly selected branch per
tree.
To test the applicability of the BNM prediction model at the
northern extreme of the central climatic zone of Maine, data
were collected from 36 branches from 12 codominant A. balsamea trees fom a 23-year-old fir--spruce stand in Bald Mountain Township, Maine during the third week of August 1993.
The study site was located approximately 100 km northwest of
Orono in the northern portion of the central climatic zone of
Maine (Briggs and Lemin 1992) and is fully described elsewhere (Newton et al. 1989, 1992).
Model formulation
Several published model forms were screened in their original
and mathematically modified forms to predict BNM, BANG,
NBW, BDIA, BL, and crown radius (CR). Models to predict
BL and CR were fit to the average measurements per whorl
collected from the tree top down to the widest portion of the
crown. Needle mass was predicted from BDIA for all whorls,
and the needle mass for all interwhorl branches was estimated
from the oven-dry/fresh interwhorl mass ratio for each tree.
Mean branch diameters (cm, ± SE) used to predict needle mass
for trees of each crown class were: OG 1.91 ± 0.033, CD 1.69
± 0.024, IT 0.97 ± 0.015, and SP 0.63 ± 0.022. Total projected
leaf area (TOTLA, m2) for each tree was calculated by multiplying the average SLA for each tree by the total estimated
needle mass per tree.
A cumulative distribution function was fit by nonlinear least
squares regression to describe the relative leaf area distribution
within the crown. For each tree, projected leaf area (PLA, m2)
was summed for each of 10 equally spaced crown sections
from the tree top down to the lowest living branch. According
to our definition of live crown base, it was possible to have
living branches below the BLC. The relative leaf area for each
crown section (LAi/TOTLA, where i = crown section) was
obtained by dividing the leaf area per crown section by TOTLA
per tree; therefore, the cumulative leaf area distribution for
each tree was equal to one.
Model selection
Model selections were based on scatterplots of residuals versus
predicted values in combination with a modified likelihood
criterion that was equal to the root mean squared error from
unweighted and untransformed equations, and to a rescaled
root mean squared error from weighted and transformed equations (Furnival 1961). Coefficients of determination (R2) were
not used as selection criteria when transformations were performed on the dependent variables. Because an adjustment for
logarithmic bias should be considered when applying results
from logarithmically transformed models (Finney 1941,
Snowdon 1991), the ratio estimator correction factor for logarithmic bias (Snowdon 1991) was applied to all estimates used
73
in subsequent analyses and in calculations of bias (bias =
(observed − predicted)/n) in the original units of measure.
Parameter estimation
Analysis of covariance (ANOCOVA, Zar 1984) was used to
test the hypothesis that a general model applicable to each
crown class would adequately describe a relationship. When
warranted, regression with indicator variables (Neter et al.
1990) was used to construct an expanded equation applicable
to each crown class. The completely expanded model can be
thought of as a combination of four equations with the parameter estimates for the CD, IT, and SP crown classes depicting
their respective differences from the OG class. The results of
these analyses can be defined by two extremes. If canopy
position had no effect on a model, a single equation would
adequately describe the relationship. Conversely, a separate
equation would be required for each crown class if pronounced
differences for a given relationship existed among canopy
positions. In this paper, only the parameter estimates in their
reduced form are presented for the models having the best fit
to describe a particular relationship. Parameter estimates, fit
statistics for the completely expanded and intermediate models, and an explicit statement of the hypotheses tested during
the construction of the final models are provided in Gilmore
(1995). All statistical analyses were done using SYSTAT
(SYSTAT, Inc. 1992, Evanston, IL).
Results
The selected models, their parameter estimates, and their respective indices of fit are presented in Table 3. The models
having best indices of fit, and thus best satisfying the assumption of least squares regression while accounting for differences in scale (Furnival 1961) are marked with an asterisk.
Branch needle mass prediction
The selected BNM prediction model was a logarithmic transformation of a model presented by Ek (1979) (Table 3). We
rejected the hypothesis (P < 0.001) that a common BNM
prediction model could be used for all crown classes. However,
comparison of the fit statistics for the general and expanded
versions of this model revealed only a small sacrifice in overall
precision for the general model compared with the expanded
model (Table 4). Therefore, because of its simplicity, we recommend Model 1 (Table 5) over the expanded model. Relative to the upper canopy positions (OG and CD), bias for the
modeling data was greatest for the prediction of needle mass
of branches from IT trees and small branches from SP trees
(Table 5, Figure 2a).
We tested the applicability of the BNM prediction model
(Model 1) constructed from data at the southern extreme of the
central climatic zone in Maine with independent data collected
at the northern extreme of this climatic zone. The BNM prediction model consistently underestimated BNM (Figure 3)
and, when we corrected for logarithmic bias, predicted values
had an average bias of 40%.
74
GILMORE AND SEYMOUR
Table 3. Models screened to predict branch needle mass, number of branches per whorl, and branch diameter. Standard errors and probability values
of parameter estimates are shown in parentheses; selected models are marked with an asterisk.
Model form1
MSE
R2
Furnival’s index
BNM = b0 + b1BDIA2
b0 = 5.979 (SE = 2.821, P = 0.034)
b1 = 21.896 (SE = 1.183, P < 0.001)
2153.091
0.459
46.40
0.798
0.748
10.22
2103.381
0.473
45.86
0.729
0.767
9.77
1960.753
0.510
44.28
1.153
0.635
12.28
2853.649
0.289
53.42
4
NBW = ln(b0) + b1ln(GI)
b0 = 4.310 (SE = 0.086, P < 0.001)
b1 = 0.834 (SE = 0.049, P < 0.001)
1.203
0.292
1.09
*5NBW = b0GI b1 [exp(b2L)]
b0 = 4.979 (SE = 0.199, P < 0.001)
b1 = 0.319 (SE = 0.021, P < 0.001)
b2 = −0.017 (SE = 0.006, P < 0.001)
1.168
0.314
1.05
Average BDIA = b1L + b2L2
b1 = 0.531 (SE = 0.011, P < 0.001)
b2 = −0.025 (SE = 0.001, P < 0.001)
0.200
0.913
0.44
*Average BDIA = b1dbh + b2L + b3L2
b1 = 0.039 (SE = 0.001, P < 0.001)
b2 = 0.301 (SE = 0.012, P < 0.001)
b3 = − 0.011 (SE = 0.001, P < 0.001)
0.104
0.955
0.32
*ln(BNM) = b0 + b1ln(BDIA)
b0 = 3.068 (SE = 0.048, P < 0.001)
b1 = 2.134 (SE = 0.062, P < 0.001)
2
BNM = b0BDIA b1
b0 = 32.443 (SE = 2.736, P < 0.001)
b1 = 1.609 (SE = 0.097, P < 0.001)
ln(BNM) = b0 + b1ln(BDIA) + b2ln(L)
b0 = 2.841 (SE = 0.059, P < 0.001)
b1 = 1.545 (SE = 0.110, P < 0.001)
b2 = 0.415 (SE = 0.066, P < 0.001)
2
BNM = b0BDIA b1Lb2
b0 = 20.773 (SE = 2.840, P < 0.001)
b1 = 1.237 (SE = 0.129, P < 0.001)
b2 = 0.487 (SE = 0.091, P < 0.001)
3
ln(BNM) = b0 + b1ln(BDIA × WA) + b2ln(NW)
b0 = 1.981 (SE = 0.467, P < 0.001)
b1 = 1.060 (SE = 0.040, P < 0.001)
b2 = −0.508 (SE = 0.160, P = 0.001)
BNM = b0(BDIA × WA)b1NW b2
b0 = 26.914 (SE = 13.280, P < 0.001)
b1 = 1.003 (SE = 0.102, P < 0.001)
b2 = −0.748 (SE = 0.202, P < 0.001)
1
2
3
4
5
BNM = branch needle mass (g), BDIA = branch basal diameter (cm), L = distance from branch base to tree top (m), H = total tree height (m),
dbh = diameter at breast height (cm, measured at 1.3 m), WA = whorl age, NW = total number of whorls in the crown.
From Ek (1979).
From Kleinschmidt et al. (1980).
From Powell (1977).
Adapted from Maguire et al. (1994).
Branch angle prediction
We explored the use of the model presented by Colin and
Houllier (1992) (Table 5), to predict BANG. Although we
rejected the hypothesis that a general model adequately de-
scribed the BANG relationship for each crown class, we present only a general model (Model 2) because, as a result of the
variation in our data and the mathematical complexity of the
model, we were not able to fit a model with indicator variables
to obtain parameters for each crown class. Predicted values for
CROWN ARCHITECTURE OF ABIES BALSAMEA
75
Table 4. Comparison of fit statistics for general prediction models applied to the pooled data for each crown class and expanded models where
parameters were calculated for each crown class. Abbreviations: BNM = branch needle mass; BANG = branch angle; BDIA = branch basal
diameter; BL = average branch length per whorl; CR = crown radius; LA/TOTLA = crown section relative leaf area; GI = previous year height
increment; dbh = diameter at breast height; L = depth in to the crown; VL as defined in Figure 1; and H = tree height.
Model
Dependent
variable
Predictor
variables
General
R2
Expanded
R2
General
MSE
Expanded
MSE
1
2
2
3
4
4
53
53
53
64
64
7
BNM
BANG average
BANG maximum
NBW
BDIA average
BDIA maximum
BL
CR average
CR maximum
CR average
CR maximum
LA/TOTLA
BDIA
Whorl age
Whorl age
GI, L
dbh, L
dbh, L
L
VL
VL
L, H
L, H
VL
0.7481
0.419
0.437
0.314
0.8161
0.7971
0.831
0.811
0.826
0.789
0.795
0.884
0.754
na2
na2
na2
0.819
0.798
0.8871
0.8601
0.8661
0.8751
0.8731
0.9111
0.7981
179.647
182.422
1.168
0.1031
0.1451
0.068
0.072
0.091
0.069
0.091
0.015
0.799
na2
na2
na2
0.102
0.144
0.0461
0.0541
0.0641
0.0491
0.0611
0.0121
1
2
3
4
Denotes the form (general or expanded) of the model selected for presentation.
An expanded model was not fit with indicator variables for reasons discussed in text.
From Mitchell (1975) model form.
From Honer (1971) model form.
Table 5. Parameter estimates and bias (expressed as a percentage of the mean for each crown class in parenthesis) among crown classes for final
models to predict branch needle mass (BNM), branch angle (BANG), the number of branches per whorl (NBW), and branch diameter (BDIA) for
Abies balsamea. Other abbreviations: WA = whorl age; and L = depth into the crown.
Model
11
Final model form
Bias
ln(BNM) = 3.068 + 2.134[ln(BDIA)]
MSE = 0.798, R2 = 0.748
Open-grown
Codominant
Intermediate
Suppressed
− 11.0 (−13.5%)
− 5.1 (−5.0%)
19.4 (42.9%)
2.5 (16.9%)
2
Average BANG = 39.296 + 51.609[1 − exp(−0.059WA)]
MSE = 179.647, R2 = 0.419
− 0.6 (−0.9%)
− 2.1 (−3.5%)
1.9 (3.0%)
1.5 (2.2%)
2
Max BANG = 43.049 + 52.404[1 − exp(− 0.064WA)]
MSE = 182.422, R2 = 0.437
− 0.6 (−0.9%)
− 1.8 (−2.7%)
1.3 (1.9%)
1.7 (2.3%)
3
NBW = 4.979GI
[exp(−0.017L)]
MSE = 1.168, R2 = 0.314
0.3 (8.3%)
0.1 (2.8%)
4
Average BDIA = 0.039dbh + 0.301L − 0.011L
MSE = 0.104, R2 = 0.816
0.2 (0.3%)
− 0.1 (−0.1%)
− 0.1 (−0.1%)
0.1 (0.1%)
4
Max BDIA = 0.041dbh + 0.349L − 0.014L
MSE = 0.145, R2 = 0.717
0.3 (14.1%)
− 0.1 (−9.0%)
− 0.1 (−5.5%)
0.1 (15.1%)
1
0.319
0 (0%)
−0.4 (− 21.5%)
2
2
Ratio correction factor for logarithmic bias = 1.144.
BANG increased as branches aged and their mass increased
with the accumulation of branchwood and needle mass (Table 5). Whorl age (WA, years) was the variable with the highest
correlation with BANG (r = 0.62, P < 0.001 and r = 0.63,
P < 0.001 with average and maximum BANG, respectively).
Other variables, such as dbh, depth into the crown (L), crown
length (CL, m), tree height (H), whorl height and branch length
(BL) had weak correlations (r < 0.48) with average or maximum BANG, and when included in various forms of the
model, their parameter estimates did not differ from zero.
Number of branches per whorl prediction
A variation of the model originally presented by Maguire et al.
(1994) to predict NBW had the best index of fit (Table 3).
Because the precision with which the model (Model 3, Table 5) predicted NBW was low (R2 = 0.314), we did not attempt
to obtain parameter estimates for each crown class with indicator variables. The absolute bias for the general NBW prediction model for each crown class was also low, however, and
bias expressed as a percentage of the mean was less than 10%
for all but the SP crown class (Table 5). Additional condition-
76
GILMORE AND SEYMOUR
Figure 3. Plotted regression line for Model 1 constructed from data
from the southern extreme of the central climatic zone of Maine and
the natural logarithm of observed branch needle mass values from
Bald Mountain Township located at the northern extreme of the central
climatic zone.
Figure 2. (A) Log-log plot of needle mass (BNM, g) versus branch
diameter (BDIA, cm) and the predictive function ln(BNM) = 3.068 +
2.134 ln(BDIA). (B) Estimates of ln(BNM) from Model 3 using
ln(BDIA) as the sole predictor versus results from the model:
ln(BNM) = 2.841 + 1.545 ln(BDIA) + 0.415 ln(L).
ing of the NBW prediction model with predictor variables to
represent tree size parameters such as dbh, H, and CL did not
improve precision.
Branch diameter prediction
We rejected the hypothesis (P < 0.001) that a common BDIA
prediction model could be used for all crown classes. Because
differences in fit statistics between the general and expanded
models were minor (Table 4), we present the general model
constructed from data pooled from all four canopy positions.
The equations constructed to predict average and maximum
BDIA from L and dbh had low bias among crown classes
(Table 5).
Branch length and crown radius prediction
We rejected the hypotheses (P < 0.001) that a common BL or
CR prediction model could be used for all crown classes. The
expanded forms of the models of Mitchell ((1975), Model 5)
and Honer ((1971), Model 6) to predict CR produced similar
R2 values and MSEs (Table 4). A comparison of the models
using the maximum CL (8 m) for a hypothetical open-grown
tree (Figure 4) indicated that crown radius estimates obtained
from Honer’s and Mitchell’s models were identical, but the
bias of Honer’s model was less (Table 6). Each of these model
forms predicted crown radii within 0.5 m of those predicted
with Honer’s (1971) original coefficients (Figure 4). Mitchell’s
model ((1975), Model 5) is simpler, requiring only one independent variable, but CR will always be predicted to increase
with increasing L because of the logarithmic transformation
incorporated into the model (Figure 5a). Honer’s model
((1971), Model 6) is more complex, requiring two independent
variables and the calculation of several interaction terms. Both
model forms behave rationally when applied to values within
the range of data from which they were constructed (Table 2,
Figures 4 and 5).
Leaf area distribution
We rejected the hypothesis (P < 0.001) that a common cumulative leaf area distribution model could be used for all crown
classes (Table 6, Figure 6a). The cumulative leaf area distribution for each tree was always one, but because there was
considerable variation in the cumulative leaf area distribution
at relative depths into the crown among trees, Model 7 never
predicted a value of one for cumulative leaf area. To visualize
the predicted relative leaf area distribution for each crown
class, Figure 6b was constructed by predicting the cumulative
leaf distribution with Model 7 (Table 6) and subtracting the
predicted leaf area distribution in the upper crown sections
from each progressively lower crown section. The leaf area
distribution in the crown at the lowest living branch, as illustrated by Figure 6b, could not be zero.
Discussion
Several studies have suggested that the inclusion of depth in
the crown (L) in a BNM prediction model improves precision
(Ek 1979, Gillespie et al. 1994). We found that BDIA and L
were highly correlated (r = 0.81, P < 0.001), and therefore, to
avoid multicollinearity, L was not included in Model 1. A
scatter plot of predicted values of ln(BNM) from Model 1
CROWN ARCHITECTURE OF ABIES BALSAMEA
Figure 4. Comparison of the fit of Honer’s (1971) and Mitchell’s
(1975) models to Honer’s (1971) published equation for Abies balsamea for a hypothetical open-grown tree of average crown length.
versus predicted values of ln(BNM) from a model incorporating ln(BDIA) and ln(L) as predictor variables revealed that
ln(L) has a minimal effect on predicted values. The greatest
77
divergence in predicted values between the two models was for
small branches from the SP crown class (Figure 2b).
Many early branch-level biomass prediction models have
not been tested with data collected outside of the geographic
location from which they were constructed (Attiwill 1962,
Loomis et al. 1966, Ek 1979). For A. balsamea, no differences
in parameter estimates among data collection points were
detected between New Brunswick, Canada and New Hampshire, USA during the construction of a branch-level biomass
prediction model (Kleinschmidt et al. 1980). However, differences in parameter estimates among research plantations were
detected for branch-level biomass prediction models constructed for Pseudotsuga menziesii (Mirb.) Franco (Kurz
1989), suggesting that branch-level biomass prediction models
should be verified with independent data collected from outside their region of construction. Our inability to validate
Model 1 across a narrow latitudinal gradient contrasts with the
results from a study of A. balsamea by Kleinschmidt et al.
(1980) who concluded that their BNM prediction model was
applicable in New Hampshire, USA and New Brunswick,
Canada. However, when applied to our data, their model form
had a poor fit relative to our BNM prediction model (Table 3).
Kleinschmidt et al. (1980) did not sample trees from the
Table 6. Parameter estimates and bias (expressed as a percentage of the mean for each crown class in parenthesis) by crown class for the final
models to predict branch length (BL) and crown radius (CR), and to describe the cumulative foliage distribution in Abies balsamea. Variable VL
is as defined in Figure 1. Other variables: H = tree height; and LAi /TOTLA = relative leaf area for each crown section.
Model
Crown class
Final model
51
Open-grown
Codominant
Intermediate
Suppressed
BL = 1.783ln[(L/3.248) + 1]
BL = 1.783ln[(L/4.023) + 1]
BL = 0.736ln[(L/1.260) + 1]
BL = 1.783ln[(L/2.258) + 1]
0.006
0.001
0.001
0.014
(0.4%)
(0.1%)
(0.1%)
(2.2%)
51
Open-grown
Codominant
Intermediate
Suppressed
Average CR = 1.763ln[(VL/3.271) + 1]
Average CR = 1.763ln[(VL/4.015) + 1]
Average CR = 0.728ln[(VL/1.193) + 1]
Average CR = 2.349ln[(VL/3.271) + 1]
0.438
0.323
0.244
0.139
(33.6%)
(32.6%)
(41.3%)
(23.1%)
51
Open-grown
Codominant
Intermediate
Suppressed
Max CR = 2.048ln[(VL/3.444) + 1]
Max CR = 2.048ln[(VL/4.187) + 1]
Max CR = 1.098ln[(VL/1.758) + 1]
Max CR = 1.098ln[(VL/0.997) + 1]
0.013
0.002
−0.123
0.124
(9.6%)
(0.2%)
(− 18.1%)
(18.3%)
62
Open-grown
Codominant
Intermediate
Suppressed
Average CR = 0.761VL − 0.025VLH − 0.604VL2/H − 0.028VL2
Average CR = 0.490VL − 0.007VLH − 0.220VL2/H − 0.003VL2
Average CR = 0.761VL − 0.025VLH − 1.665VL2/H + 0.087VL2
Average CR = 0.761VL − 0.009VLH + 0.830VL2/H − 0.182VL2
0.027
0.021
0.042
0.062
(2.1%)
(2.1%)
(7.1%)
(10.3%)
62
Open-grown
Codominant
Intermediate
Suppressed
Max CR = 0.847VL − 0.030VLH − 0.668VL2/H + 0.034VL2
Max CR = 0.391VL + 0.001VLH − 0.003VL2/H − 0.015VL2
Max CR = 0.604VL − 0.006VLH − 0.668VL2/H + 0.006VL2
Max CR = 0.847VL − 0.004VLH + 0.967VL2/H − 0.226VL2
0.019
0.019
0.017
0.047
(1.4%)
(1.8%)
(2.5%)
(6.9%)
7
Open-grown
Codominant
Intermediate
Suppressed
LAi /TOTLA = 1 − exp(− 2.748VL3.027 )
LAi /TOTLA = 1 − exp(− 3.610VL3.027 )
LAi /TOTLA = 1 − exp(− 2.748VL1.885 )
LAi /TOTLA = 1 − exp(− 2.748VL2.231 )
0.008
0.005
0.008
0.005
(0.8%)
(0.5%)
(0.8%)
(0.5%)
1
2
Model 5 = Mitchell (1975) model form.
Model 6 = Honer (1971) model form.
Bias
78
GILMORE AND SEYMOUR
Figure 5. Relative comparison of (A) Mitchell (1975) and (B) Honer
(1971) model forms among crown classes for hypothetical trees of
average crown length.
Figure 6. Predicted (A) cumulative and (B) relative leaf area distribution within the crown of Abies balsamea by crown class.
intermediate or suppressed crown classes. Kershaw (1993)
found differences in a BNM model constructed for Tsuga
heterophylla (Raf.) Sarg. among research plantations established in central Washington, USA and speculated that these
differences were attributable to differences in needle mass of
trees between regions. Herbicide trials (Newton et al. 1989,
1992) may have reduced competition enough to influence
branch density (number of branches per unit crown length) in
the trees used to test the applicability of our model to a
different region. Additionally, site quality may have influenced
the performance of the BNM prediction model outside of the
study region. Potential site productivity for A. balsamea in
Bald Mountain Township, Maine is less than that of our study
area (R.D. Briggs, personal communication). Because of the
poor performance of the BNM prediction model outside our
study area, we conclude that branch-level biomass prediction
models should be verified before widespread application.
Colin and Houllier (1992) had difficulties predicting BANG
for plantation-grown Picea abies Karst., and suggested that
unmeasured factors, such as site quality and genetics, may
influence branch angle. Although we took care to sample trees
from sites of comparable quality, sampling from a naturally
regenerated population precludes control over genetic variation among trees.
Maguire et al. (1994) included relative stand dbh as a predictor variable in their original version of Model 3 to predict
NBW to account for social position. Crown class influenced
Model 3, but parameter estimates for other variables (dbh, H)
included in the model to represent tree size did not differ from
zero. Previous-year height increments (GI, m) characteristic to
each crown class may have contributed to our inability to
expand Model 3 mathematically.
Large diameter trees were predicted to have greater BDIAs
relative to small diameter trees. Inspection of the quadratic
term for each Model 4 indicates that BDIA will eventually
decrease with depth into the crown. This is consistent with
results reported by Colin and Houllier (1992) and Maguire et
al. (1994) for plantation-grown Picea abies and Pseudotsuga
menziesii, respectively. Branch basal diameter (BDIA) growth
in the lower portion of the crowns of forest-grown trees would
be expected to slow as a result of increased mutual shading and
as branches lose their vigor and the non-foliated lower portion
of the crown becomes larger (Mitchell 1975, Jack and Long
1992). At the same time, branches in the upper portion of the
crown would be expected to retain their vigor, and their accumulated BDIA growth would eventually be greater than that in
the lower portions of the crown.
Leaf area of the understory crown classes was concentrated
in the upper 45% of the crown (Figure 6b). Morphologically,
this would increase the ability of understory trees (IT and SP)
to produce photosynthate at the low irradiances that characterize their canopy position. Crowns of SP trees were also
CROWN ARCHITECTURE OF ABIES BALSAMEA
found to be flat relative to other crown classes, with only small
tufts of foliage on their lower branches (Figure 5) and most of
their leaf area concentrated in the upper portion of the crown
(Figure 6). In contrast, leaf area of trees in the OG and CD
crown classes was concentrated in the lower 60% of the crown.
By definition (Smith 1986), OG and CD trees have adequate
growing space; therefore, one would expect older branches in
the lower portion of the crown to have greater leaf area than
younger branches higher in the crown. Crown shape morphology for each canopy position conformed to the Massart model
of crown architecture (Hallé et al. 1978). Kohyama (1980) also
observed patterns of crown architecture in OG and SP Abies
mariesii Mast. saplings that conform to the Massart model.
Measures of crown geometry can be incorporated into sophisticated growth and yield prediction models, such as The
Tree and Stand Simulator (TASS, Mitchell 1975), that use
crown size as a measure of competition, into ecosystem-based
models, such as those derived from JABOWA (Botkin et al.
1972), and into nutrient cycling models (Kimmins 1988) that
use a stratified leaf area distribution in the crown to model
growth efficiency.
Acknowledgments
Financial support for this project was provided by a grant from the
Maine Agricultural and Forest Experiment Station to R.S. Seymour
and by the University of Maine Cooperative Forestry Research Unit.
Appreciation is extended to Michael P. Auger, Kelley A. Brackley, Neil
A. Brackley, Douglas S. Campbell, Anne L. Coyle, Michael E. Day,
Christina Epperson, James Frohn, Stuart W. Gardner, Jason D. Moore,
James T. Murray, Stephen W. Murray, Francine Natale, Andrew L.
Rotch, Perry S. Sawyer, II, Brian D. Tift, David G. Ray, Dale R.
Whitlock, and Kurt E. Zschau for field and laboratory assistance. Data
from Bald Mountain Township were courtesy of Russell D. Briggs,
Richard J. Dionne, Ronald C. Lemin, Jr., and Maxwell L. McCormack, Jr. This manuscript benefited from reviews by Russell D.
Briggs, Marie R. Coyea, Michael S. Greenwood, William A. Halteman, Douglas A. Maguire, and Alan S. White. Publication 2003 of the
Maine Agricultural and Forest Experiment Station.
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