Calibrating the self-thinning frontier

Southern Cross University
ePublications@SCU
School of Environmental Science and
Management
School of Environmental Science and
Management
2009
Calibrating the self-thinning frontier
Jerome K. Vanclay
Southern Cross University, jerry.vanclay@scu.edu.au
Peter J. Sands
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Suggested Citation
Pre-print of Vanclay, JK & Sands, PJ in press, 'Calibrating the self-thinning frontier', Forest Ecology and Management, vol. 259, no. 1,
pp. 81-85.
Forest Ecology and Management home page available at www.elsevier.com/locate/foreco
Publisher's version of article available at http://dx.doi.org/10.1016/j.foreco.2009.09.045

Calibrating the self-thinning frontier

Forest Ecology and Management, in press

Jerome K. Vanclay1 and Peter J. Sands2
1 Southern Cross University, PO Box 157, Lismore NSW 2480, Australia
2 39 Oakleigh Av, Taroona, Tasmania, Australia 7053

Correspondence: JVanclay@scu.edu.au, Tel: +61 2 6620 3147, Fax: +61 2 6621 2669


Abstract
Calibration of the self-thinning frontier in even-aged monocultures is hampered by scarce
data and by subjective decisions about the proximity of data to the frontier. We present a
simple model that applies to observations of the full trajectory of stand mean diameter
across a range of densities not close to the frontier. Development of the model is based on
a consideration of the slope s = ln(Nt/Nt-1)/ln(Dt/Dt-1) of a log-transformed plot of
stocking Nt and mean stem diameter Dt at time t. This avoids the need for subjective
decisions about limiting density and allows the use of abundant data further from the self-
thinning frontier. The model can be solved analytically and yields equations for the
stocking and the stand basal area as an explicit function of stem diameter. It predicts that
self-thinning may be regulated by the maximum basal area with a slope of -2. The
significance of other predictor variables offers an effective test of competing self-thinning
theories such Yoda’s -3/2 power rule and Reineke’s stand density index.
Keywords: even-aged monoculture, maximum basal area, self-thinning, stand density


1. Introduction
The theory of limiting density (Reineke, 1933) and self-thinning (Yoda et al., 1963) in
even-aged monocultures continues to attract attention (Pretzsch, 2002; Bi, 2004; Pretzsch
and Biber, 2005; Reynolds and Ford, 2005) decades after being proposed, but an efficient
and satisfactory procedure to calibrate the self-thinning frontier remains elusive (Zhang et
al., 2005; Vanderschaaf and Burkhart, 2007). Many methods are hampered by the need to
make a subjective selection of samples considered to be representative and at or near the
frontier (Zhang et al. 2005). Despite doubts about the validity of the concept (Reynolds
and Ford, 2005), there remains a need to reduce this subjectivity because the concept is
widely applied in forest research and management.

A key principle implicit the Reineke and Yoda propositions is that any arrangement of
regular objects in a single layer within a confined area has a volume-area relationship in
which the number N of objects and their volume V exhibit a power curve V∝N-3/2
(Pretzsch, 2002) – or equivalently, that the relationship between size S and number is
N∝S-2. In a frequently cited paper, Yoda et al. (1963) observed that this Euclidean
fundamental applies to herbaceous plants. Decades earlier, Reineke (1933) observed a
slope of -1.605 in the size-stocking power curve for several north American conifers, an
observation at odds with the -2 slope indicated by Yoda’s proposition. Within a few

years, MacKinney and Chaiken (1935) completed a statistical analysis of Reineke’s
original data and estimated the slope as -1.707. More recently, Pretzsch and Biber (2005)
have argued that the slope is species-specific. West et al. (1997) have advocated a slope
of -4/3, but their analysis has been challenged (Kozlowski and Konarzewski, 2004;
Stegen and White, 2008). Many subsequent studies have examined whether these trends
do, or do not exist in plant communities (for recent reviews, see e.g., Reynolds and Ford,
2005; Shaw, 2006).

Several characteristics of the self-thinning frontier hamper empirical study and
calibration. The frontier, rather like a black hole, is not visible directly, but must be
inferred indirectly from the death of individuals as a stand approaches the frontier. The
self-thinning frontier is not a constant unyielding barrier, but is more like a water table
that fluctuates with the seasons, manifesting itself differently at times according to
limiting resources. As a result, the frontier can be estimated only indirectly,
approximately, and asymptotically.

Further complications arise from the empirical relationships that are used to describe the
frontier. Some discrepancies may arise because the space occupied by a tree is
determined in part by its crown, rather than by the stem diameter used as the basis for
Reineke’s stand density index. If the relationship between stem diameter and crown
diameter is C=βD0.8 (in the case of Reineke’s estimate), then there is no conflict, and the
stand density index complies with the expected Euclidean trend and with the crown
competition factor (Krajicek et al., 1961). Smith and Hann (1984) observed that when
there is an allometric relationship between diameter and volume, V=β D2.4, Reineke’s and
Yoda’s hypotheses concur. Recently, Zeide (2005) has suggested a modification to
Reineke’s equation to better account for tree size and packing, and Garcia (2009) has
advocated the merits of an analogous approach based on top height rather than diameter.

It can be demonstrated empirically that the slope of the number-size power curve is
unaffected by packing (i.e., regular versus random placement of trees), and by any lag
that may occur while neighbours grow into a space created by the death of a plant. Any
departure from the nominal slope of -2 is primarily due to the allometric relationship
between stem diameter and crown size, or more specifically, between stem diameter and
the space needed to satisfy photosynthetic and respiratory demands. Notwithstanding
claims by Enquist and Niklas (2001), it is reasonable to expect that trees in different
environments may exhibit different size:space relationships (Morris, 2002), influenced by
the space needed to capture limiting resources.

Yoda’s self-thinning line and Reineke’s stand density index are useful and widely used in
plantation growth models to predict natural mortality (e.g., Monserud et al., 2005),
including in process-based models (e.g., Landsberg and Waring 1997). Calibrating these
relationships is notoriously difficult and demanding of data, and this paper considers an
alternative approach to estimate self-thinning trends such as Yoda’s and Reineke’s lines.
Rather than selecting data believed to be at the self-thinning frontier, it is expedient to
examine the full trajectory of stand mean diameter across a range of densities by
examining s = ln(Nt/Nt-1)/ln(Dt/Dt-1), where Nt and Dt are the stocking and mean diameter

at time t. We present a simple model based on the assumption, supported by observations
on many stands, that s can be approximated by a power function of the current stand basal
area. The resulting model can be solved analytically to give explicit equations for both
stocking and basal area as a function of diameter. The model has two parameters: the
maximum basal area attained during self-thinning, and the power, which determines how
rapidly a stand approaches the self-thinning line. The model is very easy to fit to
observed stocking v. diameter data, and its use avoids the need for subjective decisions
about limiting density and allows the use of abundant data further from the frontier.

2. Materials and Methods
The assumption usually made in interpreting and applying the self-thinning line is that
growth slows and mortality increases as a forest stand approaches the limiting stand
density, but this assumption is rarely taken into account explicitly when estimating the
frontier. The self-thinning frontier is usually estimated by subjectively selecting data
considered to be close to the frontier, but an alternative is to examine the first differences
of successive observations of forest condition. Others (e.g., Roderick and Barnes, 2004;
Pretzsch and Biber, 2005; Zhang et al., 2005; Vanderschaaf and Burkhart, 2007) have
examined first-differences, but have not commented on the evolution of these trajectories
as they approach the frontier.

100
1000
100 1000D
N

Figure 1. Self-thinning trends in Eucalyptus pilularis forests in Queensland. High productivity
plots marked with plus symbols (+), typical plots marked with diamonds (◊), and low productivity
plots marked with crosses(×).

The slope s of the trajectory observed on the log-log graph illustrated in Fig. 1 can be
estimated as the first difference of successive observations

1
1
ln( / )(ln )
(ln ) ln( / )
t t
t t
N Nd N
s
d D D D
−
−
= ≈ , (1)
where Nt is the number of individuals and Dt is their mean size (diameter at breast height,
1.3 m) at time t. This formulation expresses the slope in the form considered by Reineke
(viz. N=f(D)), the inverse of the form considered by Yoda (V=f(N)). Note that s is not a
constant, but defines a trajectory, and is expected to have a near-zero value in stands with
low densities, and to increase and approach a limiting slope s* as density increases.
According to the Reineke and Yoda propositions, s* may be in the range -1.6 to -2. It is
useful to examine full trajectory of s across a wide range of densities because data are
often more abundant further from the self-thinning frontier, and this avoids the need for
subjective decisions about proximity to the frontier. In many cases, this approach is more
faithful to the available data, which often informs how forest stands approach the self-
thinning frontier, rather than how they behave at the frontier itself.

Table 1. Characteristics of 29 plots of Eucalyptus pilularis used to examine the self-thinning response.
Attribute Minimum Mean Maximum
Establishment date 1923 1928 1971
Stand age (years) 1 35 63
No of measures 8 19 31
Site productivity 22 32 41
Stem diameter (cm) 1 32 80
Basal area (m2/ha) 1 31 71
Stems/ha 83 365 1594

The utility of this approach was examined using data from several sources, but is
illustrated primarily with Eucalyptus pilularis Sm. data (Table 1) from a national
collection of growth and yield data from eight eucalypt species growing in even-aged,
monoculture forest (West and Mattay, 1993; Mattay and West, 1994). Plots that had not
been re- measured, and intervals involving harvesting or artificial thinning were omitted
from the analysis. Measurement intervals in these data varied greatly (3 months to 14
years), so intervals were combined to create intervals >2 years with Dt+1-Dt>1 cm to
avoid the high variance in estimates of s that may arise with pairs of observations with
minimal increment.

Death in trees may not be conspicuous and sudden. Assessors may regard a tree as ‘dead’,
only to discover green shoots at the next measure, before death is finally confirmed at
some subsequent remeasure. In addition, death is often clustered in time and space
(Vanclay, 1991a), so data derived from short intervals may exhibit a stepped approach to
the self-thinning frontier. Thus, in dense stands (>30 m2/ha), the few intervals that did not
include mortality were combined to create intervals with Nt+1<Nt. Figure 1 illustrates the
resulting data for published Eucalyptus pilularis in Queensland (Mattay and West, 1994).
In Figure 1, it is evident that the self-thinning frontier may depend on site productivity
estimated from predominant height at age 35 years (Skovsgaard and Vanclay, 2008).

Several researchers (e.g., Bi, 2001; Larsen et al., 2008; Wieskittel et al., 2009) have
previously observed that site productivity influences the self-thinning frontier.

0
1
2
3
4
0 20 40 60
-
Ln
(N
2/
N1
)/L
n
(D
2/
D1
)
Basal Area (sq m/ha)

Figure 2. Slope of the self-thinning trend s = -ln(N2/N1)/ln(D2/D1) plotted against stand basal
area for Eucalyptus pilularis in Queensland, with four lines illustrating the self-thinning
trajectories of four plots of low (▲), average (♦) and high (■) site quality

Figure 2 shows the values of s derived from this set of data, along with the actual
trajectory for s obtained from four specific stands. The large number of zero values for s
arise in part from stands that were not (yet) self-thinning, and in part from the inherent
random nature of death. However, the specific trajectories show a strong correlation with
stand basal area, and are well represented by a simple power function of basal area. We
show below that this offers a way to predict self-thinning trajectories for stands which do
not have a long history of repeated measurement. Other work (e.g., Vanclay, 1991b)
suggests that stand basal area should provide a good basis for predicting s, but other
candidates could include leaf area index (Hamilton et al., 1995; Innes et al., 2005),
aggregate height (Fei et al., 2006), or top height (Garcia, 2009). The possibility that s
may be estimated adequately from basal area alone implies that the self-thinning frontier
will have a slope s* = -2, but the inclusion of additional predictor variables such as Ln(D)
are needed to provide s*>-2 consistent with Yoda’s and Reineke’s propositions. If we
assume that s can be approximated by a power function of basal area G = piN(D/200)2
alone, i.e.
2( / )nxs G G= − (2)
then equation (1) can be integrated (see Appendix) to give explicit equations for stem
number and stand basal area as explicit functions of current stem diameter. In equation

(2), n is a power and Gx is the basal area at which s = -2 (and also the maximum basal
predicted by the model). The integrated equations of the model are

1
2
0 0
1
2
0
( ) 1 ( / )
( ) 1 ( / )
nn
x
n n
x x
N D N n D G
G D G G n D
 = + 
 = + 
(3)
where N0 is initial stocking (i.e. stocking for small D) and n0 = piN0/40000. For large D,
i.e. when self-thinning is occurring, these equations give N = (40000/pi)(Gx/D2), i.e.
N ∝ D-2, and G = Gx.

Figure 3. Results of applying equations (3) to self thinning trajectories of three stands.
(a) Eucalyptus globulus grown in Tasmania (Goodwin and Candy, 1986), with n = 4 and
Gx = 49. (b) Pinus patula grown in South Africa (Dye, 2001), with n = 6 and Gx = 57. (c)
Pinus radiata grown in South Australia (this study), with n = 3.5 and Gx = 90. The data
were fitted by eye by setting Gx and then varying n.
0
500
1000
1500
2000
2500
0 10 20 30 40
St
an
d
s
to
ck
in
g
(tr
e
es

ha
-
1 )
0
20
40
60
80
100
B
a
sa
l a
re
a


(m
2
ha
-
1)
c) Pinus radiata
0
500
1000
1500
2000
2500
0 10 20 30 40
Stem diameter (cm)
St
an
d
s
to
ck
in
g
(tr
e
es

ha
-
1 )
0
20
40
60
80
100
B
a
sa
l a
re
a


(m
2
ha
-
1)
a) Eucalyptus globulus
0
500
1000
1500
2000
2500
0 10 20 30 40
Stem diameter (cm)
St
an
d
st
o
ck
in
g
(tr
e
es

ha
-
1 )
0
20
40
60
80
100
B
as
al

a
re
a

(m
2
ha
-
1)
b) Pinus patula

3. Results
The ability of the model given by equations (3) to fit individual self-thinning trajectories
is illustrated in Figure 3 (showing the un-transformed data) for three distinct stands of
different species grown in 3 different locations. The parameters n and Gx were estimated
by fitting the model to the data by eye. The fits are fairly insensitive to the power n and in
the following analysis we apply the same power (n = 3) to a large number of stands,
although this values was not estimated in a rigorous manner.

The case of the Eucalyptus pilularis data shown in Figure 2 can be modelled with the
simple equation s = -0.436(G/H)3, where H is the expected height of predominant trees at
age 35 years. Although simple, this is an adequate model, with a small standard error
(0.021, P<0.001), and no evidence of lack of fit (P=0.35; Weisberg, 2005). Other
predictor variables such as ln(D) were not significant (P=0.2), suggesting that self-
thinning in this species is correlated with stand basal area, and that maximum basal area
(Assmann, 1970; Sterba and Monserud, 1993; Skovsgaard and Vanclay, 2008) is a
sufficient concept to explain self-thinning and offering no support for Reineke’s and
Yoda’s propositions. Figure 4 illustrates the self-thinning trend implied by this simple
equation, and confirms the adequate fit to the data.
This ability to made reasonable predictions of self-thinning by predicting the slope s from
a power of basal area was confirmed with other published (Mattay and West, 1994) and
unpublished data. Figure 5 illustrates estimates of self-thinning in Pinus radiata D.Don in
South Australia obtained for n = 3 and Gx = 95 (based on 4 plots aged 11-62 with site
quality III-IV). This suggests that the first difference approach as implemented using
equations (3) is an efficient way to estimate self-thinning in crowded stands, and that in
many cases, basal area is a useful predictor of the trajectory.
100
1000
10000
10 100D
N

Figure 4. Self-thinning trends in Eucalyptus pilularis of near-average site productivity (28-38 m
predominant height at age 35 years). Black lines are observed data (Mattay and West, 1994).
Curved horizontal lines are constructed from estimates of s with n = 3 and Gx = 1.79H where H
is the expected height of predominant trees at age 35 years. Each dash represents 2 years
growth. Diagonal dashed line represents a stand basal area of 55 m2/ha.

100
1000
10000
10 100D
N

Figure 5. Predicted and observed self-thinning of Pinus radiata in South Australia spanning an
age range 11-62 years and site quality III-IV. Dashed diagonal line is G = 95 m2/ha, and s is
estimated with n = 3 and Gx = 95.

Estimating the trajectory solely from basal area leads to a series of self thinning lines
(Figures 4 and 5) that converge toward a site-dependent maximum stand basal area with
s* = -2. It is appropriate to examine other predictor variables such as s = β0+β1G+β2lnD,
which could accommodate s* ≠ -2 and (depending on the value of β2) support the
Reineke-MacKinney proposition that s* > -2. This approach offers an efficient and non-
subjective way to estimate the slope s, and to test the adequacy of Reineke’s and Yoda’s
propositions.
4. Discussion
A slope of s* = -2 is a direct consequence of assuming s is a power function of basal area,
with no other explanatory variables, and implies that self-thinning is regulated by
maximum stand basal area rather than according to Reineke’s proposition. A self-
thinning frontier with a slope other than s* = -2 (e.g., -3/2 as proposed by Yoda) implies
that other variables additional to basal area are required to predict s*.
A slope of s* = -2 is consistent with Yoda’s proposition if V = βD3, which may apply to
some small organisms but which rarely applies to forest trees. More generally, the slope
s
*
= -2 imposes the constraint nS = -2nt where nS is the allometric power for stem volume

or mass as a function of diameter, and nt is the slope of the log-transformed stem mass v.
stand density self thinning line. If nt = -3/2, as often assumed, then nS = 3. The Reineke-
MacKinney proposition holds only if basal area is an inadequate estimator of s and
requires lnD as an additional predictor variable. Reineke’s proposition arises if s is
calibrated as s = β0+β1G-0.4lnD, but this value was not evident in the data examined.
The analysis of the full trajectory using our model appears to be a practical and efficient
way to estimate the self-thinning frontier. It minimizes the need for subjective decisions,
and allows efficient statistical testing of Yoda’s and Reineke’s propositions. This
approach suggests that in many cases, the concept of maximum stand basal area may be a
more practical and parsimonious explanation of mortality in even-aged forest
monocultures.

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Appendix: Derivation of Equations (3)

Assume that the slope s of the ln N v. ln D curve is a power function of basal area G:

(ln ) 2( / )(ln )
n
x
d N D dN
s G G
d D N dD
= = = − , (4)
where n is a power and Gx is the basal area at which the slope s is 2. The units are
assumed to be N in trees ha-1, D in cm and G in m2 ha-1, so

2
200
DG Npi  =    . (5)
Substitute (5) into (4) to get

2n nD dN N D
N dD
γ= − , (6)
where γ = 2(pi/40000Gx)n. Rearrange (6) so that N is on the left and D on the right to give
an equation that can be directly integrated using the rule that the integral of xn-1 is xn/n.
Integration gives

21
2
nN C D
n n
γ
= + , (7)
where C is the constant of integration which is set using the initial stem number, assumed
to be N0 when D = 0. The final result for stocking as a function of DBH is

1
2
0 0( ) 1 ( / ) nnxN D N n D G = +  . (8)
where n0 = piN0/40000. The corresponding basal area follows by combining (5) and (8):

( )
( )
1
22
0
2
0
1
2
0
( )
200 40000 1 /
1 ( / ) .
n n
n
x
n n
x x
N DDG D N
n D G
G G n D
pi
pi
    = =     + 
 = + 
(9)