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R E S E A R C H A R T I C L E Open Access
Linking individual-tree and whole-stand models
for forest growth and yield prediction
Quang V Cao
Abstract
Background: Different types of growth and yield models provide essential information for making informed
decisions on how to manage forests. Whole-stand models often provide well-behaved outputs at the stand level,
but lack information on stand structures. Detailed information from individual-tree models and size-class models
typically suffers from accumulation of errors. The disaggregation method, in assuming that predictions from a
whole-stand model are reliable, partitions these outputs to individual trees. On the other hand, the combination
method seeks to improve stand-level predictions from both whole-stand and individual-tree models by combining
them.
Methods: Data from 100 plots randomly selected from the Southwide Seed Source Study of loblolly pine (Pinus
taeda L.) were used to evaluate the unadjusted individual-tree model against the disaggregation and combination
methods.
Results: Compared to the whole-stand model, the combination method did not show improvements in predicting
stand attributes in this study. The combination method also did not perform as well as the disaggregation method
in tree-level predictions. The disaggregation method provided the best predictions of tree- and stand-level survival
and growth.
Conclusions: The disaggregation approach provides a link between individual-tree models and whole-stand
models, and should be considered as a better alternative to the unadjusted tree model.
Keywords: Disaggregation; Combination method; Loblolly pine; Pinus taeda
Background
Information provided by growth and yield models is
essential for forest managers to make informed decisions
on how to manage their forests. Munro (1974) classified
growth and yield models into whole-stand models and in-
dividual tree models. He further separated individual-tree
models into distance-independent and distance-dependent
models. The whole-stand models (low resolution) and
individual-tree models (high resolution) represent two ex-
tremes. In the middle are medium-resolution models such
as diameter-distribution models and stand-table projection
models, which provide information for each diameter class
(Figure 1).
Each type of model has its own benefits and draw-
backs. Whole-stand models often provide well-behaved
outputs at the stand level, but these outputs lack informa-
tion on stand structures. Detailed information from
individual-tree models and size-class models, on the other
hand, typically results in stand-level outputs that are not
as accurate or precise because they suffer from accumula-
tion of errors (Garcia 2001, Qin and Cao 2006).
Daniels and Burkhart (1988) attempted to link differ-
ent types of growth and yield models by developing a
framework for an integrated system in which models of
different resolutions are related in a unified mathemat-
ical structure. The functions used in these models can
therefore be considered invariant at different levels of
dimensionality.
Zhang et al. (1997) used the multi-response parameter
estimation developed by Bates and Watts (1987, 1988)
to constrain an individual-tree model by optimizing for
both tree and diameter-class levels. This approach was
later modified by Cao (2006) to produce a constrained
Correspondence: qcao@lsu.edu
School of Renewable Natural Resources, Louisiana State University
Agricultural Center, Baton Rouge, LA 70803, USA
© 2014 Cao; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution
License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly credited.
Cao Forest Ecosystems 2014, 1:18
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tree model that was optimized for both tree and stand
levels.
Disaggregation method is a method that has been used
by many researchers for linking an individual-tree model
and a whole-stand model (Ritchie and Hann 1997). In
this method, outputs from the individual-tree model are
adjusted such that the resulting stand summary matches
prediction from a whole-stand model.
The disaggregation method above assumes that outputs
from whole-stand models are more reliable than those
from individual-tree models. Yue et al. (2008) found that
stand-level outputs from whole-stand and individual-tree
models could be combined to improve predictions. The
weighted average approach was extended by Zhang et al.
(2010) to include stand-level outputs from a diameter dis-
tribution model.
In this paper, the disaggregation method and com-
bination method were evaluated against the unadjusted
individual-tree model by use of data from unthinned
loblolly pine (Pinus taeda L.) plantations.
Review of methods for linking individual-tree models and
whole-stand models
Stand-level summary is obtained by aggregating (or sum-
ming) tree-level outputs from individual-tree models. Be-
cause this summary is often believed to be not as accurate
and precise as direct prediction from a whole-stand
model, the individual-tree model can be adjusted such
that the resulting stand-level output matches that from a
whole-stand model. In other words, output from the
whole-stand model is disaggregated to tree level by use of
some disaggregating function.
Ritchie and Hann (1997) provided an excellent review
on disaggregation methods, classifying the disaggregat-
ing functions into additive and proportional. In the
additive growth method, the basal area growth of each
tree is equal to the average tree basal area growth plus
an adjustment based on tree basal area (Harrison and
Daniels 1988) or tree diameter (Dhote 1994). Another cat-
egory of disaggregation methods involves proportional allo-
cations that can be applied to either growth or yield. In the
proportional yield method, predicted tree basal area is
adjusted to match predicted stand basal area (Clutter and
Allison 1974, Clutter and Jones 1980, Pienaar and Harrison
1988, Nepal and Somers 1992, McTague and Stansfield
1994, 1995). The proportional growth method involves
adjusting predicted tree basal area growth to match pre-
dicted stand basal area growth (Campbell et al. 1979, Moore
et al. 1994), tree volume growth to match stand volume
growth (Dahms 1983, Zhang et al. 1993), or tree diameter
growth to match stand diameter growth (Leary et al. 1979).
Qin and Cao (2006) evaluated four methods to link an
individual-tree model and a whole-stand model by use
of disaggregation. In the proportional yield method,
the predicted tree survival probability, diameter, and
total height were multiplied by adjustment factors
(equations 1–3 of Table 1). Tree diameter and height
growth were adjusted in the proportional growth
method, while tree survival probability was adjusted based
on the ratio of dead and alive probabilities (equations
4–6 of Table 1). The constrained least squares method
(Matney et al. 1990, Cao and Baldwin 1999) was used to
adjust tree attributes (tree survival probability, squared
diameter, or total height) by minimizing the sums of
squared differences between the predicted and adjusted
attributes, subject to the constraints that the aggregations
had to match predictions from a whole-stand model
(equations 7–8 of Table 1). Finally, in the coefficient
adjustment method, adjusting coefficients were added
to modify the coefficients of the original individual-tree
model to yield stand attributes identical to those produced
by the whole-stand model (equations 10–12 of Table 1).
The four methods evaluated produced similar results,
with the coefficient adjustment selected as the method
to disaggregate predicted stand growth among trees
inthetreelist.Theadjustedtreemodelcombinedthe
low
resolution
Diameter-distribution
model
Individual-tree
model
Stand-table
p
ro
j
ection model
high
resolution
Whole-stand
model
Figure 1 Relative position of different types of growth and yield models in terms of the resolutions of the outputs.
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Table 1 List of adjustment functions used in recent methods to link models of different resolutions
Citation Method Eq. no. Adjustment function
1/
Qin and Cao (2006) Proportional yield 1 ~
p2;i¼^
p2is
^
N2
Xj
^
p2j
0
@1
A
2
~
d2
2i¼
^
d2
2i
s
^
B2=K
Xj
^
p2j
^
d2
2j
0
@1
A
3
~
h2i¼
^
h2i
s
^
V2−a
^
N2
ðÞ
bXj
^
p2j
^
d2
2j
^
h2j
0
@1
A
Proportional growth 4 ~
p2i¼
^
p2i
^
p2iþmp1−
^
p2i
ðÞ
5
~
d2
2i¼d2
1iþ
s
^
B2=K−Xj
~
p2jd2
1j
Xj
~
p2j
^
d2
2j
−d2
1j
0
B
@1
C
A
^
d2
2i
−d2
1i
6
~
h2i¼h1iþ
s
^
V2−a
^
N2
ðÞ
−bXj
~
p
2j
~
d2
2jh1j
bXj
~
p
2j
~
d2
2j
^
h2j−h1i
ðÞ
0
B
@1
C
A
^
h2i−h1i
Constrained least
squares
7~
p2i¼^
p2iþs
^
N2−X^
p2j
=n
8
~
d2
2i¼
^
d2
2j
−
~
p2iXj
~
p2j
^
d2
2j
−s
^
B2=K
Xj
~
p2
2j
0
@1
A
9
~
h2i¼
^
h2i−
~
p2i
~
d2
2iXj
~
p2j
~
d2
2j
^
h2jþsa
^
N2−
^
V2
=b
Xj
~
p2
2j
~
d4
2j
0
@1
A
Coefficient adjustment 10
~
p2i¼p1i=1þexp α0þα1H1þα2mpd1i=Dq1
ðÞ
11
~
d2i¼d1i1þexp β0þβ1lnB1þβ2A1þβ3lnH1þβ4mdd1i
Dq1
þβ5lnh1i
hino
12
~
h2i¼h1i1þexp γ0þγ1lnB1þγ2A1þγ3lnH1þγ4mhd1i
Dq1
þγ5h1i
H1
þγ6lnd1i
hin o
Cao (2006) Disaggregation 13
~
p2i¼^
pmp
2i
14 ~
d2
2i¼d2
1iþ
s
^
B2=K−
Xj
~
p2jd2
1j
Xj
~
p2j
^
d2
2j
−d2
1j
0
B
@1
C
A
^
d2
2i
−d2
1i
Constraining individual-tree
model with diameter-class
attributes
15 ^
p2i¼1=1þexp α0þα1N1þα2B1þα3d1i
½ðÞ
^
n2;k¼Xn1;k
i¼1
^
p2i
(
16
^
d2i¼d1iþβ1
A2
A1
β2
Hβ3
1Bβ4
1dβ5
1
^
b2;k¼KXn1;k
i¼1
^
p2i
^
d2
2i
8
>
<
>
:
Constraining individual-tree
model with stand attributes
17 ^
p2i¼1=1þexp α0þα1N1þα2B1þα3d1i
½ðÞ
^
N2¼Xi
^
p2i=s
(
18
^
d2i¼d1iþβ1
A2
A1
β2
Hβ3
1Bβ4
1dβ5
1
^
B2¼K
s
Xi
^
p2i
^
d2
2i
8
>
>
<
>
>
:
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best features of whole-stand and individual-tree models.
Compared to the unadjusted tree model, the adjusted
model performed better in predicting stand attributes in
terms of stand density, basal area, and volume, especially
for long projection periods. The adjusted model also
provided comparable predictions of tree diameter, height,
and survival probability.
Cao (2006) evaluated a disaggregation method against
two approaches to constrain an individual-tree model. In
the disaggregation method, the predicted tree survival
Table 1 List of adjustment functions used in recent methods to link models of different resolutions (Continued)
Yue et al. (2008) Combined estimator 19
~
B2¼w
^
B2T þ1wðÞ
^
B2S,
where wis selected to minimize the variance of
~
B2.
Zhang et al. (2010) Combined estimator 20 ~
B2¼w1
^
B2Tþw2
^
B2Sþw3
^
B2D,
where w
k
is selected to minimize XB2−
~
B2
2, and X3
kwk¼1.
Cao (2010)1 21
~
p2i¼^
pm
2i
Tree survival 222
~
p2i¼
^
p2i
^
p2iþmp1−
^
p2i
ðÞ
323
~
p2i¼1=1þexp mpα0þα3d1i
424
~
p2i¼1=1þexp α0þα1N1þα2B1þmpd1i
525
~
p2i¼^
p2iþ
s
^
N2−Xj
^
p2j
sN1−Xj
^
p2j
0
@1
A1−
^
p2i
ðÞ
Cao (2010)1 26
^
d2i¼d1iþmddβ5
1
Tree diameter growth 227
^
d2i¼d1iþβ1A2
A1
β2Hβ3
1Bβ4
1dmd
1
328
~
d2
2i¼d2
1iþ
s
^
B2=K−
Xj
~
p2jd2
1j
Xj
~
p2j
^
d2
2j
−d2
1j
0
B
@1
C
A
^
d2
2i
−d2
1i
1/
Notation:
A
1
= stand age at the beginning of the growth period.
A
2
= stand age at the end of the growth period.
H
1
= dominant height at age A
1
.
N
1
= number of trees per ha at age A
1
.
^
N2= predicted number of trees per ha at age A
2
.
B
1
= stand basal area at age A
1
.
^
B2= predicted stand basal area at age A
2
.
^
B2D= predicted stand basal area at age A
2
from a diameter distribution model.
^
B2S= predicted stand basal area at age A
2
from a whole-stand model.
^
B2T= predicted stand basal area at age A
2
from an individual-tree model.
~
B2= combined estimator for stand basal area at age A
2
.
^
V2= predicted volume per ha at age A
2
,Dq
1
= quadratic mean diameter at age A
1
.
aand b= parameters of the individual tree volume equation, vi¼aþbd2
ihi.
v
i
,d
i
, and h
i
= tree volume , dbh, and total height of tree i, respectively.
s= plot size in ha.
K=π/40 000 =constant to convert diameter in cm to area in m
2
.
n= number of trees in the plot.
d
1i
or d
1j
= dbh of tree ior jat age A
1
.
^
d2ior
^
d2j= predicted dbh of tree ior jat the end of the growth period.
~
d2i= adjusted dbh of tree iat the end of the growth period.
h
1i
= total height of tree iat age A
1
.
^
h2ior
^
h2j= predicted total height of tree ior jat the end of the growth period.
~
h2i= adjusted total height of tree iat the end of the growth period.
p
1i
= survival prob ability of tree iat age A
1
.
^
p2ior ^
p2j= predicted survival probability of tree ior jat the end of the growth period.
~
p2ior ~
p2j= adjusted survival probability of tree ior jat the end of the growth period.
α
0
…α
3
= parameters of the tree survival equation.
β
0
…β
5
= parameters of the tree diameter growth equation.
γ
0
…γ
6
= parameters of the tree height growth equation.
n
1,k
= number of trees of the k
th
diameter class at age A
1
.
^
n2;k= predicted number of trees of the k
th
diameter class at age A
2
.
^
b2;k= predicted basal area of the k
th
diameter class at age A
2
, and m
p
,m
d
, and m
h
= adjustment coefficients to be iteratively solved to ensure that the resulting
number of trees per ha, stand basal area, and stand volume, respectively, match those produced by the whole-stand model.
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probability was adjusted with a simple power function,
in which the power was iteratively solved such that the
adjusted survival probability summed up to the pre-
dicted stand density (equation 13 of Table 1). The pro-
portional growth formula was used in adjusting diameter
growth (equation 14 of Table 1). The individual-tree
model was constrained by diameter-class attributes
(equations 15–16 of Table 1) by use of the multi-
response parameter estimation method (Zhang et al.
1997, Bates and Watts 1987, 1988). Also included in the
evaluation was a similar approach to constrain the
individual-tree model by stand attributes (equations
17–18 of Table 1). Cao (2006) found that while the two
constrained models performed slightly better than the
unconstrained tree model in predicting tree and stand
attributes, the disaggregation method provided the best
predictions of tree- and stand-level survival and growth.
Cao (2010) listed different disaggregation methods for
predicting tree survival and diameter growth. These in-
clude five disaggregation methods for adjusting tree sur-
vival probability (equations 21–25 of Table 1) and three
methods for diameter growth adjustment (equations
26–28 of Table 1). His results showed that the different
methods produced similar results. Cao (2010) also found
that use of observed rather than predicted stand attri-
butes for disaggregation led to improved predictions for
tree survival and diameter growth, i.e. the quality of the
tree-level predictions in disaggregation depended on the
reliability of the stand predictions.
Yue et al. (2008) used the method introduced by Bates
and Granger (1969) and Newbold and Granger (1974) to
combine stand-level outputs from whole-stand and
individual-tree models. The combined estimator is a
weighted average of outputs from both models (equation
19 of Table 1). The optimum weights were selected to
minimize the the variance of the combined estimator.
Zhang et al. (2010) extended this approach to also in-
clude stand-level outputs from a diameter distribution
model (equation 20 of Table 1). The least-squares esti-
mate of the weights was computed according to Tang
(1992, 1994).
Methods
Stand- and tree-level growth models developed by Cao
(2006) were used in this study. The whole-stand model
consisted of equations for predicting stand density in
terms of number of trees and basal area per hectare as
follows:
^
N2;i¼N1;i=½1þexp16:3197 –42:4204 RS1;i–0:7466 H1;i
–0:0269N1;i=A1þ50:2622=A1Þ;
ð1Þ
^
B2;i¼B1;i=1þexp –3:3259–0:7800 B1;i=A1þ41:0393=A1
ð2Þ
where:
N
1,i
= number of trees per ha in plot iat age A
1
,
^
N2;i= predicted number of trees per ha in plot iat
age A
2
,
H
1,i
= average dominant and codominant height (m) of
plot iat age A
1
,
RS
1,i
= (10,000/N
1,i
)
0.5
/H
1,i
= relative spacing of plot i
at age A
1
,
B
1,i
= stand basal area (m
2
/ha) of plot iat age A
1
,and
^
B2;i= predicted stand basal area (m
2
/ha) of plot iat
age A
2
.
The individual-tree model included equations for pre-
dicting tree survival probability and diameter growth as
follows:
^
pij ¼1þexp 1:3586−0:0010N1;iþ0:1042B1;i−0:2902d1;ij
−1
ð3Þ
^
d2;ij ¼d1;ij þ0:7168 A2
A1
2:0192
H1;i
−1:0111B1;i
−0:3166d1;ij 1:5117
ð4Þ
where:
^
pij = predicted probability that tree jin plot iis alive
at age A
2
, given that it was alive at age A
1
,
d
1,ij
= diameters (cm) of tree jin plot iat age A
1
,and
^
d2;ij = predicted diameters (cm) of tree jin plot iat
age A
2
.
Data
Equations (1, 2, 3 and 4) above were derived from 100
plots from loblolly pine (Pinus taeda L.) plantations in
the Southwide Seed Source Study, which include 15 seed
sources planted at 13 locations across 10 southern states
(Wells and Wakeley 1966).
Data used in this study were from another 100 plots,
also randomly selected from the Southwide Seed Source
Study. Each 0.0164 ha plot consisted of 49 trees, planted
at a 1.8 m × 1.8 m spacing. Tree diameters and survival
were recorded at ages 10, 15, 20, and 25 years, resulting
in a total of 300 growth periods (Table 2).
Methods evaluated
In addition to the individual-tree model (equations 3
and 4), the disaggregation and combination methods
were evaluated in this study.
Disaggregation method
Thetreesurvivalprobability(
^
pij ) predicted from
equation (3) was adjusted by use of Cao’s (2010)
method as follows:
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~
pij ¼^
pij
að5Þ
where αis the adjustment coefficient used to match the
sum of adjusted tree survival probabilities (~
pij) to predic-
tions from the stand survival model (equation 1).
From equation (4), the projected tree diameter (
^
d2;ij )
was adjusted (Cao 2010) so that the resulting stand basal
area matches the prediction from the whole-stand model
(equation 2):
~
d2
2;ij ¼d2
1;ij þβ
^
d2
2;ij −d2
1;ij
ð6Þ
where:
β¼sB
b2;i=K−X~
pijd2
1;ij
X~
pij
^
d2
2;ij −d2
1;ij
hi
, and:
K=π/40 000.
Combination method
The combined estimator of stand survival was the
weighted average of stand-level predictions from the
whole-stand model (equation 1) and the individual-tree
model (equation 3). The weights were computed according
to a method described by Tang (1992, 1994) and applied
by Zhang et al. (2010). A similar procedure was applied to
compute the combined estimator for stand basal area.
Predictions from the individual-tree model were then
adjusted from the combined estimators for stand survival
and basal area, using the disaggregation method described
earlier.
Evaluation criteria
The performance of the unadjusted, disaggregation, and
combination methods was evaluated at both stand and
tree levels, based on the following statistics.
Mean difference:
MD ¼Xyi−
^
yi
ðÞ=nð7Þ
Mean absolute difference:
MAD ¼Xyi−
^
yi
jj=nð8Þ
FI ¼Xyi−
^
yi
ðÞ
2=yi−
yðÞ
2ð9Þ
Log-likelihood:
−2ln LðÞ¼−2Xpiln pi
ðÞþ
X1−pi
ðÞln 1−pi
ðÞ
hi
ð10Þ
where:
y
i
and ^
yi= observed and predicted values at the end of
the growth period of stand variables (stand survival and
basal area) or tree variables (tree diameter and survival
probability),
y= average of y
i
,
n= number of observations, and
p
i
= predicted survival probability of tree i.
Results and discussion
Table 3 shows that the whole-stand model (disaggrega-
tion method) produced the best MD and MAD values
for stand density while the combination method yielded
the best FI value. For stand basal area, all of the best
evaluation statistics came from the whole-stand model
(Table 3). At tree level, the disaggregation method
returned the best evaluation statistics for both tree sur-
vival probability and tree diameter (Table 3).
Disaggregation method
From Table 3, it is clear that the whole-stand model was
more accurate (lower MD) and precise (lower MAD and
higher FI) in predicting stand density and basal area
than the individual-tree model. The differences were
substantial. Compared to the individual-tree model, the
whole-stand model decreased MD by 88 and 97%, de-
creased MAD by 15 and 46%, and increased FI by 8 and
28% for stand density and stand basal area, respectively.
Predicted stand attributes from the tree-level model
were not as reliable because they were obtained through
summation of individual-tree predictions, resulting in
accumulation of error.
Qin and Cao (2006) showed that a tree-level model,
after being adjusted from observed stand attributes
through disaggregation, outperformed the unadjusted
tree model. They inferred that the performance of disag-
gregation models depended largely on how close the
stand predictions were to the observed values. The
whole-stand model seemed a good candidate in this
case, yielding FI values of 0.825 and 0.862 in predicting
stand density and basal area, respectively. The tree-level
statistics support this hypothesis: the disaggregation
model reduced MD by 26 and 19%, and MAD by 14 and
11% for tree survival probability and tree diameter, re-
spectively, as compared to the unadjusted tree model. It
Table 2 Means (and standard deviations) of stand and
tree attributes, by age
Attribute Stand age (years)
10 15 20 25
Dominant height (m) 9.1 (1.3) 13.4 (1.6) 16.9 (1.9) 19.9 (2.2)
Number of trees/ha 1696 (627) 1448 (548) 1143 (350) 1013 (334)
Basal area (m
2
/ha) 19.2 (5.6) 28.8 (5.9) 33.2 (8.1) 37.4 (9.4)
Tree diameter (cm) 11.6 (3.1) 15.4 (4.1) 18.7 (4.6) 21.0 (5.2)
Fit index:
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also decreased –2ln(L) for tree survival by 11% and in-
crease FI for tree diameter by 1%.
Combination method
In this study, combining stand predictions from the
whole-stand and individual-tree models resulted in pre-
dictions of stand density and basal area that were better
than those from the individual-tree model, but not as
good as those from the whole-stand model. Among six
evaluation statistics considered, the combination method
only edged the whole-stand model in fit index (0.830
versus 0.825), while came in second for the remaining
statistics. This was contrary to past reports of superior
performance by the combination method (Yue et al.
2008, Zhang et al. 2010). In a study by Zhang et al.
(2010), similar fit index values, ranging from 0.9466 to
0.9494, were obtained for predicted stand basal area
from three different types of models for the validation data
set. In this study, a considerable difference in fit index of
stand basal area prediction between the individual-tree
model (0.676) and the whole-stand model (0.862) might
result in mediocre performance of the combination
method (FI = 0.699 for stand basal area).
The tree survival model that was disaggregated from the
combined estimator gave similar evaluation statistics as
did the unadjusted tree survival equation (Table 3). On
the other hand, the tree diameter model from the combin-
ation method performed worse than the unadjusted tree
diameter growth equation (Table 3).
Tree-level predictions were disaggregated from the
whole-stand model for the disaggregation method and from
the combined estimator for the combination method. Based
on the data from this study, the disaggregation method was
better for predicting both tree survival and diameter in
terms of all evaluation statistics.
Conclusions
The disaggregation method involves adjusting outputs
from the individual-tree model to match predictions
from the whole-stand model. It was shown in previous
findings and also in this study that this method provided
better predictions of tree survival and diameter growth.
Compared to the whole-stand model, the combination
method did not show improvements in predicting stand
attributes in this study. The combination method also
did not perform as well as the disaggregation method in
tree-level predictions.
Competing interest
The author declares that he has no competing interests.
Acknowledgement
Funding for this project was provided in part by the McIntire-Stennis funds.
Received: 25 July 2014 Accepted: 4 September 2014
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Cite this article as: Cao: Linking individual-tree and whole-stand models
for forest growth and yield prediction. Forest Ecosystems 2014 1:18.
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