Development of a stand density
management diagram for evenaged pedunculate oak stands and its
use in designing thinning schedules
M. BARRIO ANTA* AND J.G. ÁLVAREZ GONZÁLEZ
Departamento de Ingeniería Agroforestal, Universidad de Santiago de Compostela, Escuela Politécnica Superior,
Campus Universitario s/n. 27002 Lugo, Spain
* Corresponding author. E-mail: [email protected]
Summary
Density management is the usual method used by silviculturists to achieve a desired future stand
condition, and one of the most effective methods of design, display and evaluation of alternative
density management regimes in even-aged stands is the use of stand density management diagrams.
In the present study, we describe a method for developing thinning schedules for even-aged
pedunculate oak (Quercus robur L.) stands in Galicia (north-western Spain), using a density
management diagram. The diagram integrates the relationships among stand density, dominant
height, quadratic mean diameter and stand volume in a single graph. The data used in its
construction were obtained from 172 sample plots located throughout Galicia. The diagram is
basically composed of two equations: the first relates the quadratic mean diameter to the stand
density and dominant height; the second relates the stand volume to the quadratic mean diameter,
stand density and dominant height. These equations were fitted simultaneously using full
information maximum likelihood. The relative spacing index is used to characterize the growing
stock and the diagram provides isolines for dominant height, number of trees per hectare, quadratic
mean diameter and stand volume. Dominant height isolines together with the site index curves allow
specification of the timing of thinnings while intermediate and final harvest volumes are calculated
using the stand volumes isolines.
Introduction
Stand density management is the process of controlling the level of growing stock through initial
spacing or subsequent thinning to realize specific
management objectives. Determination of appropriate levels of growing stock at the stand level is a
complex process involving biological, technological
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and economic factors specific to a particular management situation. The process requires selection
of upper and lower limits of growing stock, taking into account that the upper limit is chosen to
obtain acceptable stand growth and individual
tree vigour, and the lower limit is chosen to
maintain acceptable site occupancy (Dean and
Baldwin, 1996). However, the translation of
Forestry, Vol. 78, No. 3, 2005. doi:10.1093/forestry/cpi033
Advance Access publication date 23 May 2005
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FORESTRY
specific management objectives into appropriate
upper and lower levels of growing stock is the
most difficult step in designing a density management regime (Davis, 1966).
Although field trials are the best way of determining the timing of thinnings and the theoretical
limits mentioned above, they have two serious
limitations (Dean and Baldwin, 1993): they take
many years to complete and the results cannot be
applied accurately where the site quality and management objectives differ from those encountered
in the trials. An alternative approach is the use of
stand density management diagrams (SDMDs),
which are average stand-level models that graphically illustrate the relationships among yield, density and mortality through all stages of stand
development (Newton and Weetman, 1994).
The use of SDMDs is one of the most effective
methods for the design, display and evaluation of
alternative density management regimes in evenaged stands. The use of SDMDs was initially
developed by Japanese scientists in the early
1960s (e.g. Ando, 1962, 1968; Tadaki, 1963).
During the 1970s and 1980s various modifications to the original modelling approach, incorporating forest production theories, were
proposed (Newton, 1997). Nowadays, these
models have been developed for a number of
additional species and mixed stands by employing earlier modelling approaches. The diagrams
are constructed by characterizing the growing
stock, using indices that relate the average tree
size (e.g. mean weight, volume, height or diameter) to density (e.g. number of trees per hectare).
Several density indices have been used: the stand
density index proposed by Reineke (1933) (e.g.
McCarter and Long, 1986; Long et al., 1988;
Dean and Jokela, 1992; Dean and Baldwin, 1993;
Kumar et al., 1995); the self-thinning rule proposed by Yoda et al. (1963) (e.g. Drew and
Flewelling, 1977; Kim et al., 1987); or Drew and
Flewelling’s (1979) relative density index (e.g.
Flewelling et al., 1980; Newton and Weetman,
1994; Flewelling and Drew, 1985; Farnden, 1996;
Newton, 1997, 1998). One great advantage of
these density indices is that they are independent
of site quality and stand age (Long, 1985;
McCarter and Long, 1986).
The objective of the present study was to
develop a stand density management diagram for
pure, even-aged pedunculate oak (Quercus robur
L.) stands, based on the use of the relative spacing
index to characterize the growing stock. Pedunculate oak is the dominant tree species in native
forests in north-western Spain, where the climate
is wet oceanic, and it is currently the second most
common commercially grown species (occupying
more than 180 000 ha). However, fixed rotation
lengths and rigid thinning schedules, which are
sometimes inconsistent with site-specific management objectives, are characteristic features of oak
stand management. The density management diagram allows simulation of several management
regimes and the development of thinning schedules for a wide range of site qualities and management objectives.
Materials and methods
Data set
For this study a total of 172 growth and yield
plots were used. The plots were established in
monospecific natural stands throughout Galicia,
in locations with pedunculate oak present, and
were subjectively selected to represent a wide
range of site qualities, ages and stand densities.
The plot size ranged from 625 m2 to 1200 m2
depending on stand density, in order to achieve a
minimum of 50 trees per plot. The following tree
measurements were obtained within each plot:
diameter at breast height (outside bark; ±0.1 cm),
total height (±0.1 m) and age. Total plot volumes
were calculated by combining the individual
diameter and height measurements with the volume equations proposed by Barrio (2003). At the
time of sampling, mortality was nil within all
plots. In summary, a total of 172 plot measurements and associated total volume inside bark
(m3 ha− 1 ), quadratic mean diameter (cm), basal
area (m2 ha− 1 ), density (stems ha−1) and dominant
height (m) were available for analysis. Summary
statistics for the data set are shown in Table 1.
Construction of the stand density management
diagram
The stand density diagram is basically comprised
of the relative spacing index and two equations.
According to McCarter and Long (1986), a
nonlinear least-squares curve-fitting routine was
DEVELOPMENT OF A STAND DENSITY MANAGEMENT DIAGRAM
211
Table 1: Summary statistics for the data set corresponding to Quercus robur even-aged stands
Stand variable
Age (years)
Density (stems ha− 1 )
Dominant height (m)
Quadratic mean
diameter (cm)
Basal area (m2 ha− 1 )
Total volume (m3 ha− 1)
Site index (m) at
base age 50 years
Mean
Minimum
Maximum
SD
70.04
750.79
16.96
34.00
315.32
7.17
155.00
2010.8
25.58
24.98
341.35
3.33
21.41
28.22
215.81
8.60
3.41
23.94
40.00
72.88
675.98
6.10
9.17
89.99
14.00
7.30
21.91
3.28
used to develop regression models relating quadratic mean diameter, dominant height, stand density and stand volume. The first equation relates
quadratic mean diameter to stand density and
dominant height, and the second relates stand
volume to quadratic mean diameter, stand density and dominant height.
The pedunculate oak stand density management diagram has dominant height (H) and
number of stems per hectare (N) on the major
axes. Relative density or growing stock level is
represented by relative spacing index (RS)
(Wilson, 1946):
Construction of the stand density diagram
involved the follows steps:
1
Representation of the number of trees per
hectare on the y-axis and the dominant height
on the x-axis.
2 Growing stock level is expressed by the relative
spacing index (RS). The trajectories of this index
were drawn in a range of between 14–36 per
cent, representing the maximum and minimum
combination of dominant height and number of
stem per hectare observed for this species.
3 A non-linear system of two equations was
fitted
(2)
(1)
where N is the number of trees per hectare and H
dominant height. The numerator of the expression represents the average distance between
trees, assuming triangular spacing.
The relative spacing index is particularly
useful for characterizing the growing stock
levels in SDMDs for different reasons: it is
independent of site quality and stand age
except for very young stands (Schütz, 1990);
dominant height is, from a biological point of
view, the best index for establishing the thinning intervals for this species (Kenk, 1980;
Duplat, 1996), and the linkage between dominant height growth and forest production adds
further utility to these diagrams for forest management purposes.
(3)
where dg is the quadratic mean diameter (cm), N
the stand density (stems ha- 1 ), H the dominant
height (m), V the stand volume (m3 ha- 1 ) and βi,
(i = 0–5) the regression coefficients.
Equations (2) and (3) together define a structural, simultaneous equation system where N
and H are exogenous variables or variables whose
values are determined completely independently
of the system, and dg is an endogenous instrumental variable. Since there is correlation between
error components of the variables on the lefthand side and right-hand side (Borders, 1989)
the full information maximum likelihood
approach (FIML) was applied to fit both equations simultaneously using the MODEL procedure of SAS/ETS (SAS Institute, 2001).
212
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Isolines for quadratic mean diameter were
developed using equation (2) by setting dg
constant and solving for density (N) through
a range of dominant heights (H):
(4)
5 Isolines for stand volume were determined by
substituting equation (2) into equation (3)
and solving for density (N) through a range of
dominant height (H) by setting stand volume
(V) constant:
(5)
Results and discussion
Basic parameters and plot of the stand density
management diagram
Resultant coefficient estimates and regression statistics of the fitted equations (2) and (3) are shown in
Table 2. The results showed good performance for
both equations, with high precision. All the coefficients were found to be significant at a 5 per cent
level. An examination of residuals revealed that
both regression models are unbiased with respect to
the independent variables, age and site index.
An SDMD for pedunculate oak was developed
by superimposing the expected size-density trajectories using the values of relative spacing index,
the isolines for quadratic mean diameter and the
isolines for total stand volume on a bivariate
graph with dominant height on the abscissa and
number of stems per hectare on the ordinate axis
(Figure 1). The dominant height axis ranges from
8 to 30 m, while the densities range from 50 to
3000 stems per hectare on a logarithmic scale.
The relative spacing index is used to define the
thinning weight and the diagram provides isolines
for this size-density index. The uppermost line
corresponds to a value of 14 per cent, approximating the minimum relative spacing index represented in the data set. It is assumed that this
value is a reasonable approximation of the maximum size–density relationship for Quercus robur
and thus represents the maximum combination
of dominant height and number of stems per hectare possible in stands of this species in Galicia
(Díaz-Maroto, 1997; Barrio, 2003). Additional
relative spacing index lines range up to 36. The
diagram also provides lines representing constant
values of quadratic mean diameter and stand volume. Quadratic mean diameter values range from
10 to 48 cm and the isolines slope upward from
left to right, and are highly sensitive to stand density. Stand volume values range from 50 to 700
m3 ha- 1 and the isolines slope upwards, moving
from left to right in accordance with the key
principle that productivity is proportional to
dominant height growth. The range of values
Table 2: Nonlinear regression coefficients obtained by simultaneously fitting the two equation systems
predicting quadratic mean diameter (dg) and total stand volume (V) for Quercus robur even-aged stands
Equation (2)
Estimate
Asymptotic standard error
β0
β1
β2
33.6657
9.0939
-0.3307
0.0294
0.6185
0.0505
β3
β4
β5
0.000067
0.000004
1.8722
0.0352
0.9039
0.0464
Root mean square error (cm)
0.9603
R2Adj*
0.8637
Equation (3)
Estimate
Asymptotic standard error
Root mean square error (m3 ha- 1 )
11.3868
R2Adj*
0.9840
R2Adj = 1 - (ESS/CSS)(n - 1/n - p) where ESS is the error sum of squares, CSS is the corrected sum of squares,
n is the sample size and p is the number of coefficients of the model.
DEVELOPMENT OF A STAND DENSITY MANAGEMENT DIAGRAM
213
Figure 1. Stand density management diagram for Quercus robur even-aged stands in Galicia.
represented by the axes and lines were similar to
the range of values included in the data used to
construct the diagram (Table 1).
Development of a silvicultural schedule and
yield determination
Two factors determine the schedule of thinning
within the framework of the density management
diagram: the target dominant height and/or diameter at the rotation age and the lower and upper
growing stock limits. Although these criteria are
usually determined by timber objectives, they can
also be set according to any non-timber objective
that can be expressed in terms of tree size and
trees per hectare (Dean and Baldwin, 1993).
Selection of lower and upper growing stock limits often represents a silvicultural trade-off between
maximum stand growth and maximum individual
tree growth and vigour (Long, 1985). Thus, the
decision regarding appropriate levels of growing
stock will reflect stand management objectives.
The principal objective in setting the lower
growing stock limit is to maintain adequate site
occupancy. The lower limit is usually set using a
constant value of relative spacing index somewhat above canopy closure because it marks the
beginning of competition for resources. An alternative approach is to define the thinning interval
in terms of dominant height growth (e.g. Kenk,
1980; Duplat, 1996).
In practice, the upper growing stock limit for
this species may be set at a higher level than a
relative spacing index value of 20 per cent to
avoid density-related mortality and maintain an
adequate live-crown ratio for good tree vigour
(Barrio, 2003). Moreover, a thinning schedule
could be defined by combining the thinning interval, in terms of dominant height growth, with the
thinning weight defined as the increment of the
relative spacing index value that guarantees stand
stability after thinning (e.g. Pita, 1991).
The development of a stand under two different
thinning regimes is illustrated in Figure 2. In the
first (continuous line) it is assumed that the target
harvest dominant height is 28 m with a quadratic
mean diameter of 40 cm. The upper growing stock
limit is defined by a relative spacing index value of
20 per cent and the thinning intervals are based on
dominant height increments of 2 or 3 m, in accordance with the thinning schedules proposed for
forests in France and Germany (Kenk, 1980;
Duplat, 1996). The sequence of thinnings required
to reach this point is found by stair-stepping
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Figure 2. Thinning sequence for two different silvicultural schedules in the stand density management diagram for Quercus robur in Galicia. The solid line corresponds to a high-density regime and the dashed line
corresponds to a low density regime.
backwards, taking into account the upper growing
stock limit and the thinning interval.
In the second thinning regime (dashed line) it
was assumed that the target harvest was 28 m with
a quadratic mean diameter close to 50 cm. To reach
this target a thinning interval based on dominant
height increments of 3 m was combined with a
thinning weight defined as an increment of the relative spacing index of 20 per cent after thinning.
This thinning weight is similar to those proposed
by Kenk (1980) for Quercus robur in Germany.
To determine the age at which thinning and
harvesting should be carried out in a stand, site
index curves, such as those developed for this species by Barrio (2003), should be used (Figure 3).
The vertical and horizontal segments of the
stair-steps represent the thinning and postthinning phases, respectively (Figure 2). The thinning segments are drawn parallel to the y-axis on
the assumption that low thinning has no effect on
site height; the post-thinning lineal segments are
drawn parallel to the x-axis on the assumption
that there is no mortality between thinning intervals. Although density-independent mortality
may occur at any time, due to lightening strikes,
Figure 3. Dominant height curves for three different
site indexes (9, 14 and 19 m at a reference age of
50 years). Curves are derived from an equation
developed by Barrio (2003).
wind damage, frost damage, etc., it is assumed
for planning purpose that no trees are lost between
thinning, as long as the stand density remains
above a reasonable value of the relative spacing
index, e.g. 20 per cent. Similar assumptions were
adopted for the construction of SDMDs for other
DEVELOPMENT OF A STAND DENSITY MANAGEMENT DIAGRAM
species using different size–density indexes (e.g.
McCarter and Long, 1986; Dean and Baldwin,
1993; Kumar et al., 1995). However, if data from
permanent sample plots with two or more measurements are available, a mortality equation can
be constructed and included in the diagram.
Total yield can be obtained directly for any
point on the diagram using the stand volume isolines. The sum of volumes removed during each
thinning (the difference between volume before
and after thinning) and volume at the end of the
rotation represent an estimate of the total volume
produced by a specific density management
regime. Thus, it is possible to obtain directly an
estimate of the total volume for any point on the
diagram. Finally, it is possible to estimate the
merchantable volume for any top limit diameter
by applying the stand volume ratio equation (6)
developed by Barrio (2003):
)
Vi = V × exp(-0.2289 × di2.9576 × dg- 2.5654
(6)
where Vi is the stand volume (m3 ha- 1 ) to a specific top limit diameter di (cm); V is the total stand
volume (m3 ha- 1 ) and dg is the quadratic mean
diameter.
Conclusions
The diagram outlined here for developing thinning schedules for even-aged Quercus robur
stands is adaptable to a wide range of situations.
Any combination of harvest dominant height
and/or diameter and upper and lower growing
stock limits can be accommodated for a wide
range of site qualities. Because merchantable volumes can be calculated, it is relatively easy to
develop alternative thinning schedules and to
compare these alternatives using economic criteria. If one or more re-measurements are available, a mortality function can be added together
with the site index and overlaid on the diagram.
As additional information becomes available,
the diagram can be used to describe the dynamics
of managed pedunculate oak stands from various
resource management perspectives (e.g. nontimber resource production or habitat requirements of wildlife species) by overlaying this
information on the density management diagram,
thereby facilitating management decisions.
215
Although different thinning regimes can be analysed using the diagram, the lack of a mortality
submodel limits its use within the zone of imminent-competition mortality. Furthermore, silvicultural treatments can induce the so-called ‘memory
problem’(Drew and Flewelling, 1979; Long, 1985;
Jack and Long, 1996) associated with changes in
stand structure over time; i.e. a stand which was
not thinned will not have the same average size or
the same allometric relationships as a comparable
stand of the same density that was heavily thinned.
In the use of stand density management diagrams,
it is assumed that such structural differences are
either short-lived or of limited silvicultural importance (Drew and Flewelling, 1979). Therefore,
these limitations should be considered before
applying the SDMD and, according to Newton
(2003), further information about the effects of
different thinning treatments on the residual stand
structure and the temporal variation in thinning
responses is required.
Acknowledgements
The authors express their appreciation to Dr Gary Kerr
and the anonymous reviewers for their valuable suggestions. This study was financed by Xunta de Galicia,
project PGIDT99MA29101 ‘Estudio epidométrico de
las masas de Quercus robur L. en Galicia y su influencia sobre la calidad de la madera’.
References
Ando, T. 1962 Growth analysis on the natural stands of
Japanese red pine (Pinus densiflora Sieb. et. Zucc.).
II. Analysis of stand density and growth. Bulletin
No. 147. Government Forest Experiment Station,
Tokyo [In Japanese with English summary].
Ando, T. 1968 Ecological studies on the stand density
control in even-aged pure stands. Bulletin No. 210.
Government Forest Experiment Station, Tokyo [In
Japanese with English summary].
Barrio, M. 2003 Crecimiento y producción de masas
naturales de Quercus robur L. en Galicia. Ph.D. thesis, Universidad de Santiago de Compostela.
Borders, B. 1989 Systems of equations in forest stand
modelling. For. Sci. 35, 548–556.
Davis, K.P. 1966 Forest management: regulation and
valuation, 2nd edn. McGraw-Hill, New York.
Dean, J.T. and Baldwin, V.C. 1993 Using a densitymanagement diagram to develop thinning schedules
for loblolly pine plantations. Research Paper SO
216
FORESTRY
275. USDA Forest Service. Southern Forest Experimental Station.
Dean, T.J. and Baldwin, V.C. 1996 Crown management
and stand density. In Growing Trees in a Greener
World: Industrial Forestry in the 21st Century; 35th
LSU Forestry Symposium. Carter and Mason (eds).
Louisiana State University Agricultural Center, Louisiana Agricultural Experiment Station, Baton Rouge,
LA, pp. 148–159.
Dean, T.J. and Jokela, E.J. 1992 A density-management
diagram for slash pine plantations in the lower coastal
plain. South. J. Appl. For. 16, 178–185.
Díaz-Maroto, I.J. 1997 Estudio ecológico y dasométrico de las masas de carballo (Quercus robur L.)
en Galicia. Ph.D. thesis, Universidad Politécnica de
Madrid.
Drew, T.J. and Flewelling, J.W. 1977 Some recent
Japanese theories of yield–density relationships and
their application to Monterey pine plantations. For.
Sci. 23, 517–534.
Drew, T.J. and Flewelling J.W. 1979 Stand density management: an alternative approach and its application
to Douglas-fir plantations. For. Sci. 25, 518–532.
Duplat, P. 1996 Sylviculture du chêne pédunculé.
In Bulletin technique No. 31. Office National des
Forêts, pp. 15–19.
Farnden, C. 1996 Stand density management diagrams
for lodgepole pine, white spruce and interior Douglasfir. Information Report BC-X-360. Government of
Canada, Department of Natural Resources, Canadian
Forest Service, Pacific Forestry Centre, Victoria, BC .
Flewelling, J.W. and Drew, T.J. 1985 A stand density
management diagram for lodgepole pine. In Lodgepole Pine: the Species and its Management. D.M.
Baumgarter, R.G. Krebill, J.T. Arnott and G.F.
Weetman (eds). Washington State University, Pullman, WA, pp. 239–244.
Flewelling, J.W., Wiley, K.N. and Drew, T.J. 1980
Stand density management in western hemlock. Forestry Research Technical Report 042-1417 /80/32.
Weyerhauser Corporation, Western Forestry Research Centre, Centralia, WA.
Jack, S.B. and Long, J.N. 1996 Linkages between silviculture and ecology: analysis of density management
diagrams. For. Ecol. Manage. 86, 205–220.
Kenk, G.K. 1980 Werteichenproduktion und ihre
Verbesserung in Baden-Württemberg. Allgem.
Forstzeitsch. 39, 428–429.
Kim, D.K., Kim, J.W., Park, S.K., Oh, M.Y. and Yoo,
J.H. 1987 Growth analysis of natural pure young
stand of red pine in Korea and study on the deter-
mination of reasonable density. Research Reports of
the Forestry Institute No. 34. Government of Korea,
Seoul, pp. 32–40 [In Korean with English abstract].
Kumar, B.M., Long, J.N. and Kumar, P. 1995 A density
management diagram for teak plantations of Kerala
in peninsular India. For. Ecol. Manage. 74, 125–131.
Long, J.N. 1985 A practical approach to density management. For. Chron. 23, 23–26.
Long, J.N., McCarter, J.B. and Jack, S.B. 1988 A
modified density management diagram for coastal
douglas-fir. West. J. Appl. For. 3, 88–89.
McCarter, J.B. and Long, J.N. 1986 A lodgepole pine
density management diagram. West. J. Appl. For. 1,
6–11.
Newton, P.F. 1997 Stand density management diagrams: review of their development and utility in
stand-level management planning. For. Ecol. Manage. 98, 251–265.
Newton, P.F. 1998 Regional-specific algorithmic stand
density management diagram for Black Spruce.
North. J. Appl. For. 15, 94–97.
Newton, P.F. 2003 Yield prediction errors of a stand
density management program for black spruce and
consequences for model improvement. Can. J. For.
Res. 33, 490–499.
Newton, P.F. and Weetman, G.F. 1994 Stand density
management diagram for managed black spruce
stands. For. Chron. 70, 65–74.
Pita, P.A. 1991 Planes de cortas: productos intermedios.
In Seminario sobre inventario y ordenación de montes,
Vol. III. Valsaín, Segovia.
Reineke, L.H. 1933 Perfecting a stand density index for
even-aged forest. J. Agric. Res. 46, 627–638.
SAS Institute 2001 SAS/ETS™ User’s Guide, Relase
8.2, Cary, NC.
Schütz, J.P. 1990 Sylviculture. I. Princies d’education
des fôrets. Presses Polytechniques et Universitaries
Romandes, Laussanne.
Tadaki, Y. 1963 The pre-estimating of stem yield based
on the competition density effect. Bulletin No. 154.
Government Forest Experiment Station, Tokyo [In
Japanese with English summary].
Wilson, F.G. 1946 Numerical expression of stocking in
terms of height. J. For. 44, 758–761.
Yoda, K., Kira, H., Ogawa, H. and Hozumi, K. 1963
Self-thinning in overcrowded pure stands under cultivated and natural conditions. J. Biol. Osaka City
Univ. 14, 107–129.
Received 5 May 2004