Growth Equations in Forest Research: Mathematical Basis and Model Similarities

Abstract

Purpose of Review
Growth equations have been widely used in forest research, commonly to assess ecosystem-level behavior and forest management. Nevertheless, the large number of growth equations has obscured the growth-rate behavior of each of these equations and several different terms for referring to common phenomena. This review presents a unified mathematical treatment of growth-rates besides several well-known growth equations by giving their mathematical basis and representing their behavior using tree growth data as an example.

Recent Findings
We highlight the mathematical differences among several growth equations that can be better understood by using their differential equations forms rather than their integrated forms. Moreover, the assumed-and-claimed biological basis of these growth-rate models has been taken too seriously in forest research. The focus should be on using a plausible equation for the organism being modelled. We point out that more attention should be drawn to parameter estimation strategies and behavior analysis of the proposed models. Thus, it is difficult for a single model to capture all possible shapes and rates that such a complex biological process as tree growth can depict in nature.

Summary
We pointed out misleading concepts attributed to some growth equations; however, the differences come from their mathematical properties rather than pure biological reasoning. Using the tree growth data, we depict those differences. Thus, comparisons of some functional forms (at least simple ones) must be carried out before selecting a function for drawing scientific findings.

Full text

Current Forestry Reports
https://doi.org/10.1007/s40725-021-00145-8
MODELLING PRODUCTIVITY AND FUNCTION (M WATT, SECTION EDITOR)
Growth Equations in Forest Research: Mathematical Basis
and Model Similarities
Christian Salas-Eljatib1,2,3 ·Lauri Meht¨
atalo4·Timothy G. Gregoire5·Daniel P. Soto6·Rodrigo Vargas-Gaete7,8
Accepted: 24 August 2021
©The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021
Abstract
Purpose of Review Growth equations have been widely used in forest research, commonly to assess ecosystem-level
behavior and forest management. Nevertheless, the large number of growth equations has obscured the growth-rate behavior
of each of these equations and several different terms for referring to common phenomena. This review presents a unified
mathematical treatment of growth-rates besides several well-known growth equations by giving their mathematical basis
and representing their behavior using tree growth data as an example.
Recent Findings We highlight the mathematical differences among several growth equations that can be better understood
by using their differential equations forms rather than their integrated forms. Moreover, the assumed-and-claimed biological
basis of these growth-rate models has been taken too seriously in forest research. The focus should be on using a plausible
equation for the organism being modelled. We point out that more attention should be drawn to parameter estimation
strategies and behavior analysis of the proposed models. Thus, it is difficult for a single model to capture all possible shapes
and rates that such a complex biological process as tree growth can depict in nature.
Summary We pointed out misleading concepts attributed to some growth equations; however, the differences come from
their mathematical properties rather than pure biological reasoning. Using the tree growth data, we depict those differences.
Thus, comparisons of some functional forms (at least simple ones) must be carried out before selecting a function for drawing
scientific findings.
Keywords Differential equations ·Growth-rates ·von Bertalanffy ·Logistic ·Trees
Introduction
Growth is a term which everybody understands but not
necessarily in the same way. Although we can track
definitions of growth as far as Aristotle [1], one of the
first obstacles in understanding tree growth is the lack of
precision in the definition of what is meant by growth.
Growth is determined by cell division, cell extension and
cell differentiation [2].
This article is part of the Topical Collection on Modelling
Productivity and Function
Christian Salas-Eljatib
cseljatib@gmail.com
Extended author information available on the last page of the article.
Ergo, growth is a phenomenon that captures the interest
of research in a suite of disciplines [3], especially in plant
sciences [4]. The growth pattern in forest trees is divided into
primary, i.e., the growth from a bud, root, tip or another
apical meristem, and secondary, i.e., growth from the
cambium [5]. Growth implies an increase in size and the
formation of new tissue; however, it growth may occur when
older organs, particularly leaves, are dropping off faster than
new ones are being formed [6]. The term net growth includes
also this component, whereas gross growth does not. In
forestry, these terms are mainly used at the stand level,
and using volume as variable of reference, as follows: net
growth refers to the difference between volume at the end
of the period and at the beginning; gross growth must also
incorporate the growth of trees which may have died in the
period [5,7]. We refer here to the growth of variables such
as height and stem diameters as the irreversible increase in

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leading-shoot length and stem diameter, respectively [8], or
in a more physiological context, the incorporation of carbon
into structural material [2]. Nevertheless, we follow [9],
in the sense that the level of abstraction of our modelling
framework emphasizes the physically measurable exterior
tree characteristics, which we call variables.
Let us take a closer look at the use of the terms “growth”,
“increment” and “yield” in forestry. Bruce [10] stated, using
height growth equations as an example, that there are growth
and growth-rate equations, the first one being an expression
giving the variable as a function of time, and the second as
the differential of the growth equation. He also emphasized
that growth-rate is determined at one point in time. In
forestry, growth over a time period shorter than a year
may be confusing because the growth usually occurs during
summertime. Yang et al. [11] defined the term increment
as the increase in size of an organism within a certain
time interval, i.e., as the difference in observed or predicted
growth at two points in time [10]. Current annual increment
(CAI) is the difference in the growth at the beginning and
the end of the year and periodic annual increment (PA I )
for longer time units [12]. Term yield has also been used
in the same meaning as growth [12–16], but it has mainly
been associated with volume. For example, Clutter [14]
and Curtis [15] refer to growth-rate as the derivative form
and to yield as the integral of volume. Clutter [14] defined
that if integrating the growth-rate equation gives the yield
equation, the models are compatible. Even though the
forestry literature often recognizes the works by [12]and
[14] as the first to mention the consistency between growth-
rate equations and the yield equations, many researchers
in plant ecology [17–19], animal growth [20], and also in
forestry [21–23] have been aware of this relationship long
before.
In this paper, we use the following terminology that
corresponds to the mathematical explanation of derivatives
and integrals in calculus [24]: If a state variable, y,is
measured at times t0and t1,orsizeatbothtimes,wehavea
dynamic variable, therefore we also refer to them as growth
at times t0and t1(Fig. 1).
The difference between these two measurements is the
increment of that variable, for the period of time t =t1−
t0, and is symbolized by y =y1−y0. The ratio y /t ,
is referred to as the average rate of change, between t0and
t1, which is how fast the yvariable is changing, and is the
slope of a straight line connecting f(t
0)to f(t
1)in Fig. 1.
If t becomes infinitesimally small, then we are analyzing
the derivative of y,dy/dt,inyfor any particular time
(i.e., the slope of the tangent line at points t0and t1in Fig. 1),
which we call (instantaneous) growth rate. If the increment
in the state variable from t0up to a time t1,y (or dy/dt
integrated), is added to y0,thenwegety1, the growth at time
t1.
If we were able to capture the entire lifetime of most live
organisms, their empirical growth would depict a sigmoid
curve. A sigmoid or S-type curve resembles trends in the
life cycle of many living organisms and phenomena [25]
(Fig. 2a). Virtually all exterior tree dimensions develop in
a sigmoid manner with respect to time [26]. A growth
curve is typically separated in phases (Fig. 2a). Generally
three phases have been recognized [27–29], but the pattern
remains the same. Baker [28] called them acceleration,
intermediate, and deceleration and Assmann [29] youth, full
vigor, and old age; while Zedaker et al. [30], youth, maturity,
and senescence. In phase 1, yis an exponential function of
time [27]. Trees grow relatively slowly at first, increasing
their growth-rate to the point of inflection of the growth
curve [30]. In phase 2, yis directly proportional to time [27],
and in phase 3, there is a decrease of the growth-rates [27,
30].
In addition to CAI, the forestry literature recognizes
the term mean annual increment (MAI), which is found
by dividing the growth through time tby the number of
years required to produce it (red curve in Fig. 2b), i.e.,
MAI =yt/t. Growth and increment curves have long been
used for making silvicultural decisions, using as reference
the following facts. The inflection point of the CAI curve,
which occurs at the time symbolized by tain Fig. 2,isalso
the time of maximum CAI. It has been recognized as a
good opportunity to apply thinning, since the increment of
Fig. 1 Growth definitions for the state variable yobtained from
function f. Growth curve (blue), state variable at time 0 (f(t
0))and
time 1 (f(t
0)), increment (y) between t0and t1, and growth-rate
or derivative (dy/dt)att0(f(t0))andt1(f(t1)). The derivative, or
instantaneous growth-rate, is the slope of the tangent lines (black) at
points t0and t1

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Fig. 2 Growth and growth-rates
curves. aThe trajectory of the
state variable (y), generally
representing an expression of
size, is the growth curve.
Meanwhile, bgrowth-rates
curves are represented by the
current (blue) and mean (red)
annual increments. tais the time
of culmination of current annual
increment and tbis the time of
maximum mean annual
increment
residual trees is going to reach its maximum. The point tais
called the culmination of current annual increment [29–31].
The inflection point of the MAI curve, which occurs at the
time symbolized by tbin Fig. 2, represents the point where
the curves of MAI and CAI cross, and MAI is maximized.
It has been recognized as a biological criterion to choose
the rotation age of even-aged stands. More specifically, it
is the rotation that maximizes the growth. It would be the
rotation that leads to maximum carbon sequestration if the
carbon of harvested biomass would never be released back
to the atmosphere [32]. Otherwise, the optimal rotation for
carbon sequestration would be longer. Point tbwould be
an economically optimal rotation if the interest rate of the
forest owner is 0%. A higher interest rate would make the
optimal rotation shorter [33].
Growth Equations
Many empirical and theoretical growth equations have been
used in forest research. Traditionally, practitioners want to
predict the value of a random variable at time t,let’ssay
yt, as a function of some variables that might affect the
value of yt. The usual approach is to build a mathematical
model that fits the pattern of the observed data. The resulting
model is called “empirical”. On the other hand, growth
equations have been developed from biological theory about
the growth process, with parameters that have (sometimes)
biological meaning. Theoretical growth equations have
become preferred in research but not always in practice.
Some of the so called theoretical growth equations have
empirical bases. Nevertheless, we prefer to call them as such
because the parameters have some biological interpretation.
Empirical models serve a different purpose than do
theoretical models. Thereupon, empirical models should not
be viewed as poorer alternatives.
Theoretical growth equations have been developed
for many biological disciplines. Several mathematical
equations are based on theoretical constructs. Since many
growth curves are non-linear in terms of their parameters,
growth models are an important family of non-linear models
[34]. There is an extensive number of growth equations in
the literature; however, it is hard to believe that a model
with three or four parameters could describe so complicated
process as growth from birth to death [35,36]. The origin
of many non-linear models in use today can be traced to
scholarly efforts to discover laws of nature, to reveal scales
of being, and to understand the forces of life [34]. There
are several growth equations with theoretical or theoretical-
empirical bases. Here, we focus on some that have been
widely used in forestry research.

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There are many different parameterizations of a certain
growth model. Because these models start from differential
equations, many trajectories can be obtained; as a result
numerous models have been proposed based on the
differential forms. Furthermore from different differential
equations it is possible to attain the same solution or
trajectory, a mathematical fact that is not fully understood,
as expressed by [37,38]. Several authors have reported
different parameterization of growth models for forestry
and other scientific disciplines [3,34,37,39–44]. Here, we
present differential forms and solutions of selected and most
widely use models in forest science. We want to stress that
by selecting these mathematical functions, we are aiming
to compare their mathematical differences and origins, but
not providing a detailed review of all the available growth
functions that can be found.
Hereafter, all the parameters are positive, as well as time
(t) is positive, and the state variable y. We also provide
expressions of the asymptote (i.e., the maximum level for
the state variable) and the points of inflection (i.e., where
the curve changes of curvature) of all models. We shall
use Greek letters to refer to parameters and italics font
style to refer to variables. The use of the same symbols
that represent parameters in equations through the document
do not represent the mathematical equivalence, unless that
is clearly specified. For example, βor γcan be used in
different equations, but they do not imply the same number.
Monomolecular Growth Model The monomolecular model
is the following differential equation
dy
dt=β(α −y), (1)
where yis the state variable, trepresent time, and βand
αare parameters that represents the proportional constant
and final size, respectively. For this model growth is
proportional to the remaining size of the organism [34].
Therefore, the growth-rate decreases as tincreases and the
equation cannot describe sigmoidal growth. A solution of
(1) produces the growth function
yt=α1−1−y0
αe−β(t−t0).(2)
If we further assume that t0=0,y
0=0, then (2) becomes
yt=α1−e−βt,(3)
where αis the asymptote of the state variable, but the model
does not have an inflection point (Table 1). The equation
form of (3) is known as the Mitscherlich law or Mitscherlich
equation [34].
This function was proposed for height-diameter mod-
elling by [45] as well and is therefore sometimes called
Meyer’s equation in the literature.
Tab le 1 Features of the studied growth models
Model Differential form A solution Asymptote Point of inflection
Monomolecular dy
dt=β(α −y) yt=α(1−e−βt)α None
Logistic dy
dt=βy(1−y
α)y
t=α
1+eβ0−β1tαβ0
β1;α
2
Gompertz dy
dt=βy(ln α−ln y) yt=αe−e−β(t−γ) αγ;α
e
Johnson-Schumacher dy
y=βd1
t2ln yt=α−β1
teαβ
2;eα−2
Bertalanffy-Richards dy
dt=β
γyα
yγ
−1yt=α1−e−βt1
γα−ln(γ )
β;α(1−γ)
1
γ
Wei bul l dy
dt=αβ γ t γ−1e−βt yt=α(1−e−βtγ)α (γ−1
γβ )
1
γ;α(1−e
γ−1
γ)
Schnute d2y
d2t=dy
dt[−α+(1−β)γ]yt=yβ
1+(yβ
2−yβ
1)1−e−α(t−t1)
1−e−α(t2−t1)1
βαyβ
1+yβ
2−yβ
1
1−e−α(t−t1)1
βt1+t2−1
αln(β(yβ
2eαt2−yβ
1eαt1)
yβ
2−yβ
1
);(1−β)(yβ
2eαt2−yβ
1eαt1
eαt2−eαt1)
The differential form (i.e., growth-rate equation), a solution (i.e., growth equation), the asymptote and the inflection point are provided

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Logistic Growth Model The logistic growth is mostly
attributed to Verhulst’s work in 1838 [46–48]. The logistic
model imposes a restriction to the exponential growth, then
representing limiting resources for population growth.
There are many parametrizations of the logistic model,
but we shall show the one most commonly cited in the
ecological literature [46,49],
dy
dt=θy −φy2=y(θ −φy), (4)
where yis the state variable, θand φare parameters.
The differential logistic equation can also be parameter-
ized as follows [48,50],basedontreatingθas βand (θ/φ)
as α,
dy
dt=βy −β
αy2=βy 1−y
α,(5)
where the constant αhas a biological interpretation as the
carrying capacity of the environment, e.g., the maximum
population size for ecological studies or the maximum tree
size, and βis a proportionality constant.
The logistic growth equation is the simplest equation
describing sigmoidal population growth in a resource-
limited environment, and it forms the basis for many models
in ecology [48]. A solution of the differential form of the
logistic is the following growth curve,
yt=α
1+[(α−y0)/y0]e−βt ,(6)
where y0is the value of the initial condition of the state
variable y.From(5) we see that when y→α,which
happen when t→∞, the growth-rate is zero, therefore the
population being modelled or the size of any organism, does
not grow without limit. The logistic equation is symmetric
around the inflection point t0, implying that the growth rate
fulfills f(t
0−t) =f(t
0+t) for any positive t. Also notice
that, when the growth-rate of the equation (Eq. 4) is plotted
as a function of the state variable y, and not time, the point
of inflection y=α/2(Table1) is when the growth-rate
reaches its maximum.
A usual parameterization found in the forestry literature
[31,51]is
yt=α
1+eβ0−β1t(7)
which is equivalent to make (α −y0)/y0of Eq. (6)
as equal to eβ0,andβ0and β1are parameters. Robertson
[52] proposed the same differential equation (as in Eq. 4)
to describe an autocatalytic monomolecular reaction in
biochemistry. Then, the logistic equation is also known
as the expression that represents the “autocatalytic law”
of physiology and chemistry. When [53] commented on
some recent studies on growth including the one by
[52], further generalized this thought to brain (and other)
growth. According to [34], Pearl and Reed promoted the
autocatalytic concept not only for individual but also for
population growth, in their work of 1924. In sociology, the
logistic model is know as the law of Verhulst established
in 1838 describing the growth of human populations with
limited resources [54].
Gompertz Growth Model This growth model was proposed
by [55], who was concerned with modelling mortality in
an arithmetical progression; however, [56] derived it as a
growth model. Gompertz’s differential model is
dlny
dt=β(ln α−ln y),
1
y
dy
dt=β(ln α−ln y),
dy
dt=βy(ln α−ln y), (8)
when the relative growth-rate declines with lnyand the
other terms are parameters.
The most common parameterization of a solution of (8)
is [3,34]
yt=αe−e−β(t−γ),(9)
where αis the asymptote and γis the time where the point
of inflection occurs (Table 1). Although βis a dimensionless
parameter, it affects the maximum growth-rate (i.e., αβ/e).
As opposed to the logistic model, the Gompertz curve is
asymmetric.
Johnson-Schumacher Growth Model This model was inde-
pendently proposed by [57]and[21], which has been also
known as the “reciprocal function”.
Schumacher [21] explained the model by saying that
the growth percent varies inversely with age, which he
expressed in a differential form as follows:
dy
y=βd1
t. (10)
A solution to this differential equation is found by first, set
u=t−1and then substitute du=−t−2dtfor d(1/t) =
−t−2dt in Eq. (10)
dy
y=βdu,
1
ydy=βdu,
ln y+γ1=βu +γ2,(11)
where γ1and γ2are the constants of integration. Second, by
assigning α=γ2−γ1, we arrive at
ln y=α+β1
t. (12)
This model has been widely used in forestry, mainly for
fitting height-age models. Equation (12) allows parameter
estimation in the context of linear modelling. However, one
should notice that model fitting for ln yleads to biased

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predictions of y[58,59]. Also the alternative form y=
αexp(β
t), is used in forest sciences; however, here αhas a
different interpretation than in (12).
The differential equation of [21] can be represented as the
growth-rate as a fraction of the state variable y, as follows
⎛
⎝dy
d1
t⎞
⎠/y =β. (13)
however, as mentioned by [60], this differential equation
does not represent what [21] said in words, i.e., “growth
per cent varies inversely with age”. Instead, that thought is
represented by the following differential equation
dy
dt/y =β1
t. (14)
A solution of Eq. (14) is obtained as follows,
dy
y=β1
tdt,
1
ydy=β1
tdt,
ln y+γ1=βln t+γ2,(15)
where γ1and γ2are the constants of integration, and α=
γ2−γ1, as before, we get
ln y=α+βln t, (16)
This function is not the growth equation proposed by
[21], which is being advocated to represent growth-rate
as a percentage inversely proportional to time. Using
exponential transformation, we see that the equation is a
linearized version of the extensively used power function
y=φtα,whereφ=eα.
Furnival [60] also detected the error in [21], and said “the
curve given by [21] is obtained if growth percentage is taken
as inversely proportional to the square of age”. If we proceed
as above, we arrive to
ln y=α+β−1
t. (17)
This formulation emphasizes that model (12) leads to
increasing logarithmic growth only if β<0, whereas in
(17) we need to have β>0. Therefore, we favor this
expression as a correct solution of the differential equation
dy
y=βd1
t2. (18)
Finally, we could say that the derivation of Schumacher
is correct but his formulation is wrong.
Bertanlanffy Growth Model This model was first published
in German [61], and later the author introduced it to the
English literature in a brief paper [62], and in a more
developed one [20].
He originally developed it for weight growth of animals.
The model is a result of a deeper developed basic idea of
P¨utter in 1920 [54]. That is, growth can be considered a
result of a counteraction of synthesis and destruction, of the
anabolism and catabolism of the building materials of the
body. The differential form of the von Bertanlanffy model is
dy
dt=ηyθ−κyφ,(19)
where ηand κare constants of anabolism and catabolism
respectively, and the exponents θand φindicate that both
are proportional to some power of the state variable y. Leary
[63]termed(19) as the Bertalanffy’s anabolic-catabolic
balance equation of biological growth. As explained by
[20], this differential equation represents “the change of
yis given by the difference between the processes of
building up and breaking down”. Nonetheless, this is a
property that is also shared by the monomolecular (1),
logistic (4)andGompertz(8) equations, in the sense that the
differential equation can be separated into different additive
terms. In the Bertalanffy model, both these processes are
described by the simple power models, but the form of these
submodels is not justified by biological theory. Ricker [47]
pointed out that von Bertalanffy did not make clear what
he meant by anabolism and catabolism, and aside from
all the widely quoted references to his model as a solid
theoretical growth model, that is not actually true. For
instance, von Bertalanffy [54] explained, when referring
to his model, that the use of the exponent θand φin
(19) has a pure empirical base, since it is well known that
the size dependence of physiological processes can well
be approximated by allometric expressions. More recently,
Seber and Wild [3] noticed that unfortunately the biological
basis of these models (referring to the logistic, Gompertz,
and von Bertalanffy) has been taken too seriously and had
led to the use of ill-fitting growth curves.
Bertalanffy [20] justified that φ=1 based on
physiological experience; hence
dy
dt=ηyθ−κy, (20)
and he also empirically found that for a wide class of
animals, the allometric power for the metabolic rate (θ)is
2/3. Thus, differential equation (19) becomes
dy
dt=ηy2/3−κy. (21)
Equation (21) is usually termed the von Bertalanffy
model [3]; nonetheless, we prefer to reserve that denomina-
tion to Eq. (20).
Bertalanffy-Richards Growth Model Richards [19] doubted
some theoretical considerations of the Bertalanffy model
(21), and treated some of the parameters to be related to a

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point of inflection [3]. The Richard’s differential equation is
dy
dt=β
γyα
yγ
−1,(22)
where α, β and γare the asymptote, a scale, and shape-
related parameters [34]. Although, this equation has been
termed as the Bertalanffy-Richards model by [64,65],
and also as a generalization of the Bertanlanffy model
[19], it is paramount to understand that it is simply a
reparametrization of (20) as shown by [38]. A solution of
this differential equation is
yt=α1−1−y0
αγe−β(t−t0)1
γ. (23)
Notice that (23) becomes the following growth equation if
y0=t0=0,
yt=α1−e−βt1
γ. (24)
Hence, (24) is a special case of (23), being the former
suitable for forestry plantations and the latter for natural
forests [66].
It is key to realize, and as aforementioned, that from
distinct differential equations, we can arrive at a same
solution. Notice that from the following linear differential
equation with a power transformation [64],
dyγ
dt=βαγ−yγ,(25)
we can obtain the same solution of (23), as explained in
detail by [38].
Chapman [67] starting from the von Bertalanffy differ-
ential equation, in the same way as [19] did, derived a new
parameterization of the model particularly suitable for fish-
eries population modelling. Because of that, sometimes this
equation has been called “Chapman-Richards” as well [40,
68]. We can argue that the denomination for the general-
ized differential equation (20) should be credited to [20],
because even though he did not state it formally in his paper,
he used it as the basis to obtain his proposed solution. Nev-
ertheless, Richards [19] made this clearer, and also gave his
famous 2/3 exponent for θin (20). Thus, the denomination
“Bertalanffy-Richards” seems a better compromise. Any
other denomination, e.g., Chapman-Richards, even though
correct in the sense that they started from the differen-
tial equation (20), is burdensome. Similar claims had been
brought by [37–39,69]. Based on the shortcomings men-
tioned above, we shall focus on the Bertalanffy-Richards
growth model (23).
Weibull Growth Model This model is a probability density
function (pdf) widely used in engineering [70]aswellas
in forestry for describing diameter distributions [71,72]
because of its versatility. Prodan [73] noticed that tree
increment and growth curves resemble a probability density
(i.e., frequency distribution) and a probability distribution
(i.e., cumulative frequencies distribution), respectively.
Later, Yeng et al. [11] exploited this relationship and used
it to fit growth curves of trees. Based on the review given
by [74], it seems that [11] were the first to notice this. The
differential form of the Weibull growth model is as follows
[40]:
dy
dt=αβ γ t γ−1e−βtγ. (26)
A solution of the Weibull growth-rate model is [11,41]
yt=α1−e−βtγ,(27)
where α,β,andγare parameters representing the
asymptote, intrinsic rate and the shape of the curve,
respectively. The last two parameters affects the point of
inflection (Table 1). Zeide [40,41] and Payandeh and
Wang [ 75] found that the Weibull growth model performed
as well as other common growth equations in modelling tree
growth of several variables.
Schnute Growth Model Schnute [76] introduced a model
first tested for fisheries research. The model is capable of
assuming a wide range of characteristic shapes that describe
asymptotic as well as non-asymptotic growth trends [65,
77]. The Schnute’s model is based upon acceleration in
growth, that is the rate of a rate, hence using second
derivatives as follows [76]:
d2y
d2t=dy
dt[−α+(1−β)1
y],(28)
where α,β,anddy/dtare parameters. A solution of the
Schnute differential equation is [76]
yt=τβ
1+(τ β
2−τβ
1)1−e−α(t−t1)
1−e−α(t2−t1)1/β
,(29)
where α= 0, β= 0, tis the time of interest, t1time
at beginning (e.g., young), t2time at end of (e.g., old),
τ1and τ2are parameters representing the state variable y
at time t1and t2, respectively. As stated by [65], φis a
constant acceleration in growth-rate, and βis incremental
acceleration. Unfortunately, the growth function (29) does
not have a single parameter with biological interpretation
(Table 1). The Schnute growth-rate model is among the very
few growth investigations based upon second derivatives,
with the exception of the lucid study of [78] when modelling
height growth.
The Use of Growth Equations in Empirical
Studies
Although the logistic model has been long used in
population dynamics in ecology, it has barely been used in
forestry to model growth [51]. Among them, Cooper [22],

Curr Forestry Rep
MacKinney and Schumacher [31], and Monserud [51]used
it for modelling growth of stand volume, stem area, and tree
height growth, respectively.
Unlike the logistic model, the Gompertz model is not
symmetric about the point of inflection.
It is a model more used in ecology than in forestry; it
performed well in population models as reported by the
comparison of [79] and was used with success by [80]. In
forestry, it was first mentioned by [81], but was not used
until the work of [82] in modelling stand volume growth.
The Bertalanffy-Richards model is used more than any
other function in studies at tree and stand level [41,43].
This equation was first used in forestry by [83], but
mostly popularized by [68], after which the model was
termed and later known as “Chapman-Richards”.
The Schumacher growth equation (17) is long-established
for modelling height growth of dominant trees, because it is
linear in its parameters. Notwithstanding, lately the Bertalanffy-
Richards model has become the preferred model.
The Weibull model was first proposed by [11]. Pro-
dan [73] earlier had mentioned the resemblance of growth
and increment curves to probability density and probability
distribution.
The Schnute model has the property of having asymptotic
as well as non-asymptotic shape depending on the parameter
values combination, therefore being versatile to a wider
ranges of situations. Bredenkamp and Gregoire [77]were
the first to introduce the model to forestry. Later on, Yuncai
et al. [65,84]and[85] used this model as well.
Comparisons among growth equations are valuable both
for practical and theoretical use; nonetheless, it seems that
researchers tend to prefer one of the models beforehand.
Researchers might conduct initial screenings, but may
not report the results in detail. Considering the large number
of growth functions [43], very few studies have empirically
compared them. In Table 2, we have summarized a
comprehensive list of studies in this regard. Overall, the
authors selected their models using traditional goodness-of-
fit statistics. The Bertalanffy-Richards, Weibull and Schnute
were the most widely selected according to these criteria.
Practical Demonstration
To illustrate the different growth-rates, we estimated the
parameters of the growth equations by using time series
height data of 107 Nothofagus dombeyi sample trees, a
native evergreen tree species of South America temperate
forests [93], growing on secondary stands in the south of the
country. The tree-level data were collected within 30 sample
plots established throughout the ecological distribution of
the species. Sample trees, at the time of sampling, ranged
from young ages to mature (between 21 and 71 years), as
well as from small trees to taller ones (between 10 and 34
m, Table 3).
Tab le 2 Empirical comparisons of growth equations in some forestry studies
Model
Source Monomolecular Logistic Gompertz Bertalanffy-Richards Weibull Schnute Other Data
[11]•• • √Tree vo lume
[86]√••Stand volume
[77]•√Tree diameter
[40]•• • • √Tree diameter
[75]•√Tree height
[87]••• √Plant disease
[65]√√
Tree diameter
[88]•√√√
Tree height
[84]√√
Tree height
[89]•• √√√
Tree height
[43]•••• √Tree height
[90]åTree basal a rea
[91]•√•• ••Weight
[85]åTree height
[92]√•√Plant disease
The symbol •and √represents that the equation was assessed and selected as the best in the study, respectively

Curr Forestry Rep
Tab le 3 Tree level variables summary for the 107 Nothofagus dombeyi
trees at the time of sampling. dis diameter at breast height, his total
height, age is total age, and bha is breast-height age
Statistic dhagebha
(cm) (m) (yr) (yr)
Minimum 5.3 9.9 21 15
Maximum 60.2 33.7 71 68
Mean 26.6 19.9 41.3 37.7
Median 26 20.8 40 37
CV(%) 19 16.6 15.8 17.3
The height growth data cover a wide variety of growing
conditions (Fig. 3). Further details on the data can be found
in [38].
The growth equations were fitted in a mixed-effects
model framework for considering the grouped structure of
thedata[66,94].
A general non-linear mixed-effects model for the kth
observation on the jth tree at the ith plot is
yij k =f(φij ,t
ij k )+εij k ,(30)
where fis a growth equation and the parameter vector φij
includes the parameters of the selected growth function.
Each parameter consists of a fixed effect (μ)andtwo
random effects, i.e., plot (bi)andtree(bij ), therefore
φij =μ+bi+bij . (31)
Fig. 3 Tree height growth series of dominant trees for Nothofagus
dombeyi in southern Chile. Gray lines join successive observations of
height on the same tree
We assume that the random effects among plots are inde-
pendent with bj∼N(0,D), as well as for the random
effects among trees within plots with bij ∼N(0,E).Also
the residual errors are independent and normally distributed
with Var(εij k )=σ2. Further details on the formulation
of non-linear mixed-effects models are presented in [59,
95–97].
The models were fitted using the nlme package [98]of
R[99].
The fixed-effects parameter estimates for each growth
model are displayed in Table 4. We also computed
the prediction statistics as in [100], i.e., the root mean
square differences (RMSD) and the aggregated difference
(AD) using the population level predicted value from the
corresponding mixed-effects fitted model. These statistics
provide some insight into model fit. Except for the Johnson-
Schmacher equation, all models have similar precision and
accuracy levels.
Using the fixed-effects estimated parameters of the non-
linear mixed-effects growth models (Table 4), we represent
the height growth or trajectory of the state variable h
(Fig. 4a), and the instantaneous growth-rates versus time
(Fig. 4b) and versus the state variable (Fig. 4c), as well.
These trajectories should be interpreted as the estimates for
a typical sample plot of the data set.
The number of parameters (i.e., coefficients) affects the
flexibility of functions. As such, the greater number of
parameters, the greater flexibility or intrinsic curvature.
We used two functions (i.e., Monomolecular and Johnson-
Schumacher) having 2 parameters, four (i.e., logistic,
Gomperts, Bertlanffy-Richards, and Weibull) having 3
parameters, and the Schnute’s function having 4 parameters.
Comparing flexibility of the functions for a given
number of parameters is of great interest when a generally
applicable function is searched for empirical growth
modelling. See [101] for such an analysis in the context
of diameter distribution models. However, we leave such
analysis for future studies.
Caveats
In physics, it is rare to analyze different models because they
are already well established, and uncertainty is not a central
feature. On the contrary, in biology-related disciplines, such
as forestry, variability and uncertainty is the rule. As such,
selecting a growth equation on which our findings are based
is not straightforward. Most practitioners might compare a
suite of equations and choose one of the merit of goodness-
of-fit indices [102,103]. On the contrary, others will start
with a preferred growth equation at the very beginning of
their studies [104–106].
In this vein, it is often not recognized the tremendous
impact that a selected growth model will have on our

Curr Forestry Rep
Tab le 4 Fixed-effects parameter estimates and prediction statis-
tics for each mixed-effects growth model. All models were fitted
by maximum-likelihood and using time series height data of 107
Nothofagus dombeyi sample trees from southern Chile. RMSD is the
root mean square differences and AD the aggregated difference
Parameters RMSD AD
Growth model α
βγ(m) % (m) %
Monomolecular 49.76 0.012 – 2.8249 26.2 0.0507 0.47
Logistic 23.41 2.242 0.10 2.8560 26.5 0.0689 0.63
Gompertz 26.28 0.057 18.59 2.8003 25.9 0.1082 1.00
Johnson-Schumacher 2.72 5.681 – 5.1607 47.9 1.6846 15.62
Bertalanffy-Richards 34.67 0.024 0.81 2.7876 25.9 0.1494 1.39
Weibull 32.70 0.012 1.18 2.7871 25.8 0.1487 1.38
τ1τ2
Schnute 0.16 -2.13 2.84 18.02 2.9419 27.3 0.2120 1.96
inferences. For instance, let us assume that the goodness-
of-fit indices are similar among the suite of growth
equations examined here as a reference, a pattern that is
not unrealistic. However, if we are interested in finding
the maximum height for the species, a critical functional
trait, the asymptote of the growth curves (Fig. 4a) varies
tremendously between an unrealistic value of 15.2 m for
the Johnson-Schumacher model (we used the [58]bias
correction for prediction) and 49.8 m for the monomolecular
one (Table 4).
Overall, there is no theoretical reason to limit either tree or
forest growth analysis by using a single equation, and claims
on different reparametrizations have been proposed [107].
As previously, most growth models are non-linear. The
only linear model (in the parameters) assessed here was the
Johnson-Schumacher (Eq. 17). There is no closed formula
to estimate non-linear models’ parameters, as we have
for the linear models. Nowadays, most statistical software
includes tools for easing the fit of non-linear equations, such
as self-starting functions that find good initial guesses for
the parameters [97]. However, a little more care and time
may be needed than with linear models, e.g., to run the
numerical estimation routines in large data sets, to find good
enough initial guesses for the parameters, and to ensure
that the estimates found are the global maximum of the
likelihood function, but not local [3].
Regardless, several attempts have been devoted to
ecological applications to offer guidance for fitting non-
linear growth models, since [108,109]to[44], highlighting
that this is still an important topic when modelling tree
and forest growth. Consequently, the possibility of working
with models having parameters with biological or physical
interpretation is an advantage for fitting these non-linear
models.
There are numerous applications for the unified mathe-
matical treatment of growth equations presented here. The
Fig. 4 Growth (a) and growth-rates (b) versus time, and growth-rates versus the state variable cfor the growth equations studied here

Curr Forestry Rep
more obvious are in forest modelling, such as the ones fully
described in [110,111]. Besides, growth equations have sev-
eral implications in decision support and scientific findings
[112]. Nonetheless, there are a handful of other applica-
tions where growth equations are used. Among them, we
can mention the studies focus on: ecophysiological aspects
of sapling growth [104–106,113]; forest-dynamics [114];
silviculture [115,116]; and restoration ecology [117–119],
where all the findings are based upon a single fitted–and–
assessed growth function.
As lucidly summarized by [3], we can use x(i.e., another
state variable) instead of tin the growth-rate equations, and
by this simple twirl, the same mathematical models shown
here, as well as their implications, can serve for representing
tree allometry [120], a major scientific endeavor in biology
[121,122]. In the forestry context, growth equations have
been used in modelling height-diameter [88,123,124], and
stem taper [125,126] allometric relationships.
Growth-rates of the tree and forest growing processes
are essential in research [23,127,128], especially to study
the partitioning of components, resources, and allocation
of them to different processes. At least a basic but correct
mathematical understanding of differential equations is
paramount for researching these complex processes. Similar
advocacy has been claimed by others in ecology [129].
It is noteworthy to realize how the forestry literature is
way more focused on growth than on the growth-rates
themselves; that is to say, forestry is a discipline where we
seem to be trained to think in an integrated form rather
than in a differential form. In contrast, we highlight the
pioneer works on differential equations in forestry led by
[64,130–133].
Although we might think that the reasons for focusing
on growth, but not on growth-rates when modelling forestry
variables are a handful; the main one could be related
to the fact that instantaneous growth rates are essentially
unobservable, whereas realized sizes can be measured and
so are amenable to exploratory data analysis. However, this
can also be applied to other biological disciplines, such
as wildlife ecology and landscape ecology. Regardless, the
use of growth-rates, or differential equations, are just a
few compared to size-time equations (i.e., a differential
equation solution). Maybe the reasons are also related to a
weak educational background in calculus of ecology-related
scientists [129,134]. Accordantly, we favor using growth-
rates functions, such as differential equations, in forest
growth modelling when justified by the research question
[23,38,128,133].
As a reflection of the above, the analyses of the
growth rates is mandatory. The growth-rates against time
for each model broadly vary (Fig. 4b). All of them
implies completely different processes. For instance, the
Monomolecular and the Johnson-Schumacher curves seem
unrealistic. Both the Bertalanffy-Richards and the Weibull
functions behave quite similar at reaching their maximum
growth-rates at much earlier stages. On the contrary, the
logistic, Gompertz and Bertalanffy-Richards seems to be
more feasible. Finally, a graphic representing the growth-
rates against time (Fig. 4b) is usually the most commonly
reported; however, plotting the growth-rates against size
(Fig. 4c) is also essential.
These two figures describe completely different pro-
cesses. For instance, the logistic growth-rate is symmetric,
and the point on which this occurs is where half of the
asymptote is reached, and the growth-rate is maximum.
Nonetheless, no biological reasons for the symmetry have
been presented. The asymmetric behavior of the other func-
tions seems more realistic. Both the Bertalanffy-Richards
and the Weibull behave quite similarly, reaching their max-
imum growth-rates at much smaller sizes. Finally, the
Monomolecular and the Johnson-Schumacher curves are
entirely unrealistic.
In general, the rules on which the growth equations are
based are rather broad and motivated more by mathematical
formulation than by a solid, established theory about the
biological processes of growth.
Concluding Remarks
We have shown that the mathematical differences among
several growth equations can be more easily explained
and understood using their differential equation than
their integrated forms. Inter alia, the assumed-and-claimed
biological basis of these growth-rate models has been
taken too seriously, and the focus should be on using a
plausible equation for the organism being modelled. More
attention should be put on parameter estimation strategies
and behavior analysis of the proposed models. It is difficult
for a single function to capture all possible shapes and rates
that such a complex biological process as tree growth can
depict in nature. Therefore, comparisons of some functional
forms (at least simple ones) must be carried out before
selecting a function for drawing scientific findings.
Funding This study was supported by the Chilean research grants
Fondecyt No. 1191816 and FONDEF No. ID19|10421, Academy of
Finland Flagship Programme (UNITE, decision number 337655), and
ANID BASAL FB210015.
Declarations
Conflict of Interest The authors declare that they have no conflict of
interest
.
Human and Animal Rights and Informed Consent This article does not
contain any studies with human or animal subjects performed by the
authors.

Curr Forestry Rep
Papers of particular interest, published recently, have been
highlighted as:
•Of importance
•• Of major importance
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Affiliations
Christian Salas-Eljatib1,2,3 ·Lauri Meht¨
atalo4·Timothy G. Gregoire5·Daniel P. Soto6·Rodrigo Vargas-Gaete7,8
1Centro de Modelaci´
on y Monitoreo de Ecosistemas, Universidad
Mayor, Santiago, Chile
2Vicerrector´
ıa de Investigaci´
on y Postgrado, Universidad de La
Frontera, Temuco, Chile
3Departamento de Silvicultura y Conservaci´
on de la Naturaleza,
Universidad de Chile, Santiago, Chile
4Natural Resources Institute Finland (Luke), Joensuu, Finland
5School of the Environment, Yale University,
New Haven, CT, USA
6Departamento de Ciencias Naturales y Tecnolog´
ıa, Universidad de
Ay s ´
en, Coyhaique, Chile
7Laboratorio de Biometr´
ıa, Departamento de Ciencias Forestales,
Universidad de La Frontera, Temuco, Chile
8Centro Nacional de Excelencia para la Industria de la Madera
(CENAMAD), Pontificia Universidad Cat´
olica de Chile,
Santiago, Chile