Directory UMM :Data Elmu:jurnal:T:Tree Physiology:Vol15.1995:

Tree Physiology 15, 113--120
© 1995 Heron Publishing----Victoria, Canada

Prediction of stem profile of Picea abies using a process-based tree
growth model
CHRISTINE DELEUZE1 and FRANÇOIS HOULLIER2
1 Laboratoire de Biométrie, Génétique et Biologie des Populations, CNRS-URA 243, Université Claude Bernard Lyon I, 43 Boulevard du 11 novembre
1918, 69622 Villeurbanne Cedex, France
2

Unité Dynamique des Systèmes Forestiers, ENGREF, 14 rue Girardet, 54042 Nancy Cedex, France

Received April 12, 1994

Summary We built a simple tree growth model for Norway
spruce (Picea abies (L.) Karst.) that describes the biomass and
stem radial growth of one tree in a stand. Growth is controlled
by an external height growth function that accounts for site
quality. Crown recession is represented by an empirical function that accounts for the limitation to crown development
caused by mechanical contacts with neighboring trees. The
model describes biomass growth based on carbon budget (photosynthesis, respiration and senescence) and carbon partitioning between foliage, stem and root compartments. An internal

regulation is introduced based on a functional balance between
crown and root development. Stem annual growth is distributed
along the stem by means of an empirical rule. Stem profile is
the final output of the model and can be used to check the
overall consistency of the model and as an aid in wood quality
studies. The underlying assumptions of the model are described.
Keywords: carbon budget, process model, stem analysis.

Introduction
Most forest growth and yield models are empirical and are
based on a statistical description of tree or stand growth (Houllier et al. 1991). Their validity is therefore limited to a particular range of climatic, geographic and silvicultural conditions.
Because these conditions vary from one region to another and
are likely to change over time, it is necessary to update and
calibrate these empirical models regularly. Such calibration
usually requires generating large data sets from permanent
plots and so there is a long delay before the models can be
updated. It seems essential, therefore, to develop new, more
biologically oriented models whose structure will reduce, or
even avoid, the need for (and the cost of) periodic calibration
procedures (Kimmins 1990, Sievänen 1993).

Process-based models, the result of a well-developed modeling approach, are often based on a detailed description of
physiological processes, thus they are complex and mostly
restricted to research or educational applications (Landsberg et
al. 1991, Thornley 1991). We propose an alternative approach

intermediate between knowledge-based and empirically based
modeling strategies. Our objective was to derive a forest management-oriented model from general biological principles
that can be fitted to empirical field data. Thus, our goal was not
to describe tree growth processes in detail but to build a general
model that approximates tree growth and can be extrapolated
to varying growth conditions.
Most process-based growth models predict biomass growth
(Reynolds and Thornley 1982, McMurtrie 1985, Mäkelä 1986)
but do not provide a detailed description of the stem. Because
radial growth is readily measured and because tree profile is of
prime importance for timber quality (Väisänen et al. 1989), our
model was designed to describe both radial and height growth
and to predict the internal ring structure of the stem. One
problem with this a priori choice is that the links between
primary and secondary growth in the whole plant involve

mechanisms and regulations that are not well understood. As
stated by Sloboda and Pfreundt (1989), height growth is one of
the unsolved problems in process-based growth modeling. To
avoid this difficulty, Sievänen et al. (1987, 1988) and Sievänen
and Burk (1991) used an allometric relationship between
height and radial dimensions; Valentine (1985) and Mäkelä
(1986) applied the pipe model theory; and Sloboda and Pfreundt (1989) used an external height growth function.
We have built a single-tree model that is controlled by an
empirical site-dependent height growth curve, which was fitted to field data, and by a mechanical constraint to crown
development. The height growth function can be regarded as
an expression of site quality and genetic control (cf. empirical
forest growth models). The underlying assumptions of the
model are explicitly stated and discussed in the context of
potential applications and further refinements. We also compare the predictions of the model with field growth data.

Data
Site
Data were obtained from an unthinned stand of pure, evenaged Norway spruce (Picea abies (L.) Karst.) located on a flat

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DELEUZE AND HOULLIER

area at Moncel-sur-Seille, near Nancy (northeastern France).
The height of the dominant trees in the stand at 50 years was
26.9 m. The stand was 53 years old in 1991 when six trees (two
dominant, two codominant and two suppressed trees; Houllier
1993) were selected, felled and measured.

trie 1991). Foliage and stem are described by both dry weight
and geometrical dimensions.
Photosynthesis and carbon assimilation
The annual dry weight increment depends on annual leaf
photosynthetic production (P), which is given by:

Measurements
For each tree, we measured total height (H), diameter at breast
height (D), crown length (Lc) and the radius of the horizontal
projection of the crown along eight directions (Rc). Height
growth was described throughout the life of the tree, based on

the scars located at the top of each growth unit. The angle,
length and horizontal extension of one branch whorl per
growth unit were measured to describe crown morphology.
Eleven or 12 disks were then cut from each tree to obtain a
description of annual increment along the stem.

Model structure
The model (Figure 1) is recurrent, that is, the state of the tree
at date t determines the growth between t and t + 1 and, hence,
the state of the tree at date t + 1.
Description of the tree
The tree is regarded as a simple functional system with three
compartments: foliage, stem and roots (cf. McMurtrie 1985,
Sievänen et al. 1987, 1988, Sievänen and Burk 1991, McMur-

P = σ c Wf ,

(1)

where σc is specific photosynthetic activity and Wf denotes leaf

biomass, which we assumed was correlated with current-year
leaf area, Sf, by the equation: Wf = df Sf, where df is leaf weight
per unit area. We used Sf because (i) it is directly related to the
area that intercepts light, and (ii) leaves are approximately
structured in successive shells corresponding to each year
(Mitchell 1975). Miller (1986) suggests that the growth of
evergreens is correlated more closely with the amount of
foliage located in the external shell (current-year foliage) than
with the total amount of foliage, because nutrients are translocated from older to younger leaves and older leaves have a
lower photosynthetic efficiency than younger leaves.
We assumed that net assimilation (∆W), which is the difference between photosynthesis and growth respiration (R), is
proportional to growth (with a constant γ). Thus where
∆W = P − R and R = γ∆W:
∆W =

P
.
1+γ

(2)

Figure 1. General structure of the model.
Dotted lines account for recursive processes from year t to year t + 1.

PREDICTION OF STEM PROFILE USING A PROCESS-MODEL

Carbon partitioning
Net assimilation (∆W) is partitioned between foliage, stem and
roots by means of allocation coefficients: λf for foliage, λs for
stem and λr for roots. Conservation of mass in this system
implies that:
λ f + λr + λs = 1 .

(3)

For each compartment there is a biomass balance: net
growth = growth allocation − turnover − maintenance respiration, where turnover is the proportion (si) of living biomass that
is lost by senescence, and maintenance respiration is the respiration necessary to maintain living biomass.
For a stand model, stem senescence (ss) signifies the death
of some trees, but for this single tree model, the tree either dies

or survives; therefore we did not take senescence of the stem
into account and set ss = 0. For root and foliage, maintenance
respiration is proportional (with a constant ri) to foliage
biomass (Wf) and root biomass (Wr). However, for the stem,
maintenance respiration is correlated more with stem area (Ss)
than with stem biomass (Ws) (Kinerson 1975):
∆W f = λf ∆W − (r f + sf)Wf,

(4)

∆W r = λr ∆W − (r r + sr)Wr,

(5)

∆ W s = λs ∆ W − r s S s .

(6)

Functional balance
We used the principle of functional balance (Reynolds and

Thornley 1982, Mäkelä 1986) to link the photosynthetic activity of foliage to the assimilation of nutrients by roots. We
supposed implicitly that (i) carbon and nutrients are used in a
constant ratio for dry matter growth (this should be true for a
large range of environmental conditions), (ii) assimilation of
elements is correlated with their utilization, and (iii) storage of
carbon and nutrients is constant in the tree during the time of
simulation (these reserves are affected by environmental perturbations, and this first model could be improved by the
introduction of a storage compartment that would play the role
of a buffer). The principle of functional balance assumes that
the quantity of assimilated carbon (σcWf) is proportional to the
uptake of nutrients by roots (σnWr):
π n σ c W f = σ n W r,

(7)

where σn is the specific assimilation coefficient and πn is a
constant.
Resolution: external height growth function and crown
shape function
At this stage, the model contains eight state variables (P, ∆W,

Wf, Wr, Ws, λf, γr, γs) but only seven equations. Moreover, we

115

need a geometrical description of foliar and stem compartments to estimate Sf and Ss.
The system could be solved either by introducing another
constraint or another equation (e.g., using a hypothesis of
growth optimization (McMurtrie 1985, 1991), or another
physiological process, for example the pipe model assumption
(Mäkelä 1986, Valentine 1985, 1987)), or by decreasing the
number of state variables (e.g., setting the allocation coefficients to constant values (McMurtrie 1985)).
A second possibility would be to link W and S by an allometric relationship between biomass and area (the pipe model is
such a relation (Shinozaki et al. 1964)) or between biomass and
height (there are such empirical relations for stem but not for
foliage: e.g., Ws = ν D2H (Sievänen and Burk 1991)).
In principle, the pipe model theory solves both problems.
However, we did not choose this solution because the range of
conditions where it can be applied is not well defined. Moreover the relationship between foliar mass and cross-sectional
area at the base of the crown is altered by environmental
variations (Granier 1981, Aussenac et al. 1981, Long and

Smith 1984, Dean and Long 1986, Long and Smith 1988,
Aussenac and Granier 1988), and may also be altered by
thinning. We chose, therefore, to use an external description for
height growth and crown development.
The height function is the Chapman-Richards equation:
b

H (t) = b11 − exp (−b2t) ,


 3

(8)

where t is time (age), b1 is the upper asymptote, b2 is a
parameter related to maximum height growth (b1b2 is the
growth rate at beginning), and b3 is a form parameter that was
set to 2.
We supposed that the shape of the crown does not change
over the life of a tree and compared two simple equations: a
cone (Equation 9) and a logarithmic function (Equation 10)
proposed by Mitchell (1975) and Ottorini (1991):
B(h,t) = a(H(t) − h),


H(t) − h 
,
B(h,t) = a1a2 ln 1 +
a3 


(9)

(10)

where h is the vertical position along the stem (0 < h < H),
H − h is the distance to the leader, and B(h,t) is the branch
extension at height h and age t.
In a closed stand, crown recession is determined by mechanical and space-based tree-to-tree competition (Mitchell
1975, Ottorini 1991). Because we simulate only one tree, and
do not have a detailed description of its neighbor trees in the
stand, we used an empirical crown recession function (Figure 2):
∆H b = f(H )∆H

where

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DELEUZE AND HOULLIER

f(H ) =

c1
c3

 c2 
1+ 
H

(11)

,

and c1 is the asymptotic value of the crown recession/height
growth ratio, c2 is approximately the height where crown
recession starts, and c3 is the shape parameter.
Equations 8, 9 or 10, and 11 allow the calculation of Wf for
each year. The system can then be solved and ∆Ws determined.
Stem growth is finally partitioned along the stem according to
Pressler’s empirical rule (Figure 3), which is similar to the pipe
model theory: ‘‘the area increment on any part of the stem is
proportional to the foliage capacity in the upper part of the tree,
and therefore is nearly equal in all parts of the stem, which are
free from branches’’ (Assmann 1970).
Fitting the empirical external functions

Figure 3. Pressler’s rule: left is the cumulated (from top downward)
foliar area (or mass), and right is the stem basal area increment.

sponds to a constant angle of 11.17°, and Equation 10 was
fitted, tree by tree, to crown data (Figure 4).
Use of Pressler’s rule as a method for estimating the base of
the functional part of the crown

Height function
The Chapman-Richards equation was adjusted tree by tree
with a nonlinear procedure (Table 1 and Figure 2) using a
differential mode, i.e., annual height growth was adjusted
instead of cumulated height.
Crown shape
The parameter a of Equation 9 was set to 0.197, which corre-

Figure 2. Adjusted height growth curve (upper line) on height data
(points) and theoretical crown recession (lower line) compared with
adjusted crown base (points) using Pressler’s rule. Example for
Tree a72.

Our tree model requires information about the successive positions of the base of the crown. The height to the base of the
crown is not a usual measurement: in our data set we only knew
its position in 1991. Moreover, it has not been established
whether the theoretical base of the functional crown is directly
related to the field measurements of the first living branch, the
first living whorl or the first whorl free of any contacts with

Figure 4. Adjusted crown shape (line) on crown data (points) for Tree
a72 (Equation 10): x-axis is the distance from leader (m), and y-axis is
the crown extension (m).

Table 1. Estimated parameters for Equations 8 and 10. Abbreviations: N = number of experimental points, SSE = sum of squared errors, fixed
denotes that this parameter was not estimated but fixed.1
Tree no.

b1

b2

b3

N

SSE

a1

a2

a3

N

SSE

a72
a88
a94
a124
a170
a201

45.2431
39.5563
39.5255
44.3679
40.2753
44.6615

0.03271
0.03666
0.03774
0.03542
0.03678
0.03345

2 (fixed)
2 (fixed)
2 (fixed)
2 (fixed)
2 (fixed)
2 (fixed)

45
45
45
45
45
45

0.7638
1.6000
1.3167
1.6013
2.3114
2.2653

0.828 (fixed)
0.828 (fixed)
0.828 (fixed)
0.828 (fixed)
0.828 (fixed)
0.828 (fixed)

1.1570
1.2690
2.0761
1.3231
2.3691
3.4941

1.7884
1.8926
3.6428
2.0625
4.4032
7.6095

17
14
10
15
12
13

0.7079
0.6757
0.2848
1.3647
1.2928
0.3601

1

We used the software ‘‘multilisa’’ developed in our laboratory by Jean Christophe Hervé.

PREDICTION OF STEM PROFILE USING A PROCESS-MODEL

neighbor trees (Colin and Houllier 1992). Therefore, to obtain
information about the functional part of the crown and to
determine the parameters for our crown recession function
(Equation 11), we used Pressler’s rule.
Stem analysis provided an estimate of ∆as(h), the cross-sectional area growth of the stem at various heights h. According
to Pressler’s rule, ∆as(h) is constant below the crown base and
decreases above the crown base to ∆as(H) = 0. If we assume
that the shape of the crown is conical, Pressler’s rule leads to a
simple segmented model: the first segment (below the functional part of the crown) corresponds to a constant value, and
the second segment (within the functional part of the crown)
decreases linearly with crown height (Figure 5). We thus estimated the successive positions of the base of the functional
crown by the following procedure. The stem analysis data were
used to adjust the two linear segments for each date, the point
where the two segments intersected being an estimate of the
height of the base of the crown for each tree (see Figure 5).
Simulations
Parameters and initial conditions
Physiological parameters were obtained from the literature.
Because Norway spruce is rarely studied in process models,
some parameters were taken from other species (Table 2).The
initial values of the model state variables were obtained from
stem analysis data at age 20.
Simulation of internal stem structure
To simulate the model, a program was written in C++. Twelve
simulations were done: two for each tree, that is one with a
conical crown and one with the logarithmic function of crown
shape. Crown and stem were assumed to be symmetric around
the vertical axis and were geometrically defined by their horizontal radius at any height.
The program produces a schematic growth figure similar to
a ‘‘herring-bone’’ (Figure 6). On the left side of Figure 6, each
line represents the profile of the crown at one date; on the right

Figure 5. Adjustment of Pressler’s rule (example for Tree a170, ring
1972): x-axis is the height in the tree (cm), and y-axis is the stem area
increment (cm2). The intersection between the two segments provides
an estimate of the theoretical basis of the efficient part of the crown.

117

side of Figure 6, fine lines represent simulated stem profiles at
each date, and thick lines represent the same but from observed
stem analysis data at ages 25, 30, 40 and 50 years (simulation
begins at age 20).
Radial growth at breast height versus date was drawn for
each simulation, and compared with the observed stem analysis data (Figure 7).

Results and discussion
Qualitative validation: stem profile
For a conical crown, the simulated stem profile had a marked
inflexion point at the top of the stem, a feature that does not
appear in the data. This inflexion point was less evident when
we used the logarithmic function for crown shape. Because the
actual crown shape lies between the conical and logarithmic
form, it is likely that the existence of this inflexion point in the
simulation outputs was due to the Pressler’s rule which seems
to be invalid in the top part of the stem. If Pressler’s rule, and
Huber’s value (xylem cross-sectional area per foliar biomass
above height h (Ewers and Zimmermann 1984)) are brought
together, we find that the ratio of stem cross-sectional area
increment/foliage biomass above is not constant and increases
in the top part of the stem.
Quantitative validation
The comparison of simulated and actual diameter growth
curves provided a first test of the model (see Figure 7). The
intermediate parameters such as partitioning coefficients allowed an internal control of the model performance. Figure 8
shows a progressive stabilization of the coefficients to values
(λf ≈ 0.2, λr ≈ 0.2, λs ≈ 0.6), which were comparable to those
obtained in other process models (Mäkelä 1986, McMurtrie
1991) and to proportions given by Cannell (1985, 1989).
Improvement and perspectives
The model could be improved by better descriptions of foliage
compartment (crown) and crown-to-stem relationships
(Pressler’s rule) and by replacement of the analytic conic and
logarithmic functions of crown profile by a true dynamic
model that would relate branch growth to height growth and to
distance to the tree top (Mitchell 1975). We could also replace
the area of the crown, Sf, by the volume of the external shell of
foliage, which includes the youngest leaves.
To study the effects of environmental changes, we could
improve the functional balance with the introduction of a
storage compartment. This would in turn require another equation to solve the system. Because the utilization of carbon and
nutrients is linked to water use in the tree, the models could
also be improved by the introduction of a water compartment.
The procedure used to assess the parameters of the model
could be improved. Because our objective was to build a
soundly based model rather than to make precise and accurate
predictions, we focused on the qualitative behavior of the
model rather than on its statistical properties. To obtain a more
rigorous statistical evaluation of this complex model, we need

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DELEUZE AND HOULLIER

Table 2. Summary of all of the parameters of the model.
Name

Meaning

Unit

Value

Wf
Sf
df
Wr
Ws
Ss
ρ
P
∆W
R
γ
λi
σc

Foliage biomass
Foliage area
Specific foliage area density
Root biomass
Stem biomass
Stem external area
Specific weight
Annual photosynthetic production
Total annual growth
Growth respiration
Growth respiration coefficient
Partitioning coefficient of compartment i
Specific photosynthetic activity

kgDW
m2
kgDWm −2
kgDW
kgDW
m2
kgDWm −3
kgDW
kgDW
kgDW
unitless
unitless
unitless

Variable
Variable
0.17
Variable
Variable
Variable
390
Variable
Variable
Variable
0.23
Variable
2.7

σn
πn
rf, rr
rs
sf
sr
H(t)
h
∆sa(h)
b1
b2
b3
c1
c2
c3
a
a1
a2
a3

Specific root assimilation activity
Constant of functional balance
Maintenance respiration coefficient of foliage and root
Maintenance respiration coefficient of stem
Senescence coefficient of foliage
Senescence coefficient of roots
Total height at t
Height in the tree
Wood area increment at height h
Asymptote of height growth
Height growth parameter: maximum height growth
Height growth parameter: shape of the curve
Asymptotic value of the ratio ∆Hb:∆H
Height where crown recession starts
Form parameter
Constant for conical crown
First parameter for logarithmic crown
Second parameter for logarithmic crown
Third parameter for logarithmic crown

unitless
unitless
unitless
kg m −2
unitless
unitless
m
m
m2
m
year −1
unitless
unitless
m −1
unitless
unitless
unitless
m
m

0.04
0.01
0.1
0.01
0.2
0.5
Variable
Variable
Variable
see Table 1
see Table 1
see Table 1
0.78
10
6
0.1974
see Table 1
see Table 1
see Table 1

Source

Ceulemans and Saugier 1991, Granier 1981

Assmann 1970, Pardé 1980, Cannell 1989

Mäkelä 1986, Cannell 1989
Ceulemans and Saugier 1991,
Saugier and Garcia de Cortazar 1991
Mäkelä 1986
Mäkelä 1986
Mäkelä 1986
Mäkelä 1986
Mäkelä 1986, Cannell 1989
Mäkelä 1986, Cannell 1989

Figure 6. Graphical output of the
program. On the left, lines represent the successive simulated
crown profiles (one line per
date). On the right, fine lines represent the successive simulated
stem profiles (one line per date),
and thick lines represent observed stem analysis data at ages
25, 30, 40 and 50 years (simulation begins at age 20). Example
for Tree a72 using (a) conical
and (b) logarithmic crown profiles.

PREDICTION OF STEM PROFILE USING A PROCESS-MODEL

Figure 7. Evolution of simulated stem radius at breast height: x-axis is
the age (in years), and y-axis is the stem radius (in mm). Example for
Tree a72: ✕ = real data, n = simulated radius with conical crown, and
s = simulated radius with logarithmic crown.

119

The model requires only an external height growth function
(which describes site and genetic control) and physiological
parameters (which could be extrapolated to any similar stand).
The model provides a qualitative description of tree growth
conditional on a few parameters (site index and crown recession due to competition). Moreover, the simulation of the inner
ring structure of the stem gives indications about wood quality,
because wood density is closely linked to ring width in Norway spruce (Nepveu et al. 1988). The inner ring structure could
be introduced in wood quality simulation software (e.g., the
‘‘SIMQUA’’ software developed by Leban and Duchanois
1990).
The model could also be incorporated in a stand growth
model where the external crown recession function could be
replaced by a dynamic process describing the tree-to-tree
crown contacts. Building such a tree-and-stand model would
require an analysis of the qualitative behavior of the model at
the tree and stand levels, and a quantitative comparison of
simulated growth to observed data at both levels (Sievänen and
Burk 1991).
Acknowledgments
The authors are grateful to J.C. Hervé, who assisted with the fitting
procedure run on software ‘‘Multilisa’’ and to D. Rittié and M. Ravart
for field measurements.
References

Figure 8. Evolution of partitioning coefficients to stem and foliage for
Tree a72. Partition coefficents: λf (j for the conical crown and d for
the logarithmic crown) and λs (h for the conical crown and s for the
logarithmic crown).

to use a global fitting procedure (Sievänen et al. 1987). The
sensitivity of the model also needs to be assessed (e.g., with
Monte-Carlo simulations, see Mäkelä 1988).

Conclusion
The model provides a simple framework that can serve as a
basis for other process models: it allows us to check the
consistency of global and simple physiological hypotheses
with dendrometrical data (stem analysis). The assumptions of
the model are clearly defined, so that its range of validity can
be stated and it is possible to modify the model for a particular
aim.

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