Improving estimations of maximal stand density by combining Reineke’s size-density rule and the yield level, using the example of spruce (Picea abies (L.) Karst.) and European Beech (Fagus sylvatica L.)
Ann. For. Sci.
Improving estimations of maximal stand density by combining Reineke's size-density rule and the yield level, using the example of spruce (Picea abies (L.) Karst.) and European Beech (Fagus sylvatica L.)
• The Reineke Stand density rule relating stem numbers to the quadratic mean diameter is generally used as a reference for modelling maximal stand density. • The linearity of this relationship after double logarithmic transformation is generally assumed, but it must be questioned for untouched stands and stands with a conventional thinning regime. Curvilinearity is demonstrated for some spruce and beech stands in Switzerland and shown to be statistically representative. This relationship is independent of the site index. It can be interpreted as a change in mortality in young stages mainly due to competition and in older stages more to ageing. • A more accurate estimation of the maximal stand density needs to take into account the important variation around the mean course, known as the yield level. A simple method to assess the yield level of any stand regardless of whether it is thinned or not is presented, based on estimating the effect of a stand opening on the basal area.
Reineke / maximal stand density / yield level / carrying capacity / growth model / mortality
densité maximale /
niveau de production /
potentiel de production /
modèle de croissance /
Jean-Philippe Shc utz ¨
*, Andreas Zing
WSL CH 8093 Birmensdorf, Switzerland
The search for an objective and more or less invariant, but
comprehensible, determination of stand density has
preoccupied forest scientists for a long time. It belongs to
conventional wisdom on forest yield to use therefore the so called
stand density rule or self thinning rule of Reineke
, relying on the size to density relationship N vs. dg (the
quadratic mean diameter at breast height). Reineke considered
this rule to have a general character not only in forestry. Such
an overall applicable size-density rule is, however, too simple
to be biologically credible
. Its general
application to different tree species has been repeatedly refuted
(Lonsdale, 1990; Weller, 1987; White, 1981; Zeide, 1985)
because each species, depending on its competitive tolerance,
can exhibit other closure features. Even the applicability of
the rule to the same tree species during stand development has
been questioned because, in accordance with Zeide (1987):
“Horizontal canopy closure of even-aged stands is in
constant change; it is a process. The degree of horizontal crown
closure is affected by two opposing and mutually related
factors: lateral growth of crowns, which increases crown closure,
and mortality, which decreases it.” Thus the linearity of N
vs. dg in a double logarithmic grid has been questioned.
presented results from long-term yield series from
south Germany and demonstrated log(N) to log(dg)
relationship was, at least in some cases, non-linear.
One of the problems regarding the change in the steepness
of the so called self-thinning line in the log grid leading to a
convex curvilinear form is the question of a clear definition
of what full stand closure is. In ageing stands, instability
induced by long-term overcrowding leads to mortality, which
originally affects only suppressed low social trees in juvenile
stages (density-dependent mortality), affecting higher social
elements from the adult stage on (active mortality due to
hazardous events and ageing). It can result in changes in canopy
closure which, if not excessive, are compensated for by the
lateral growth of neighbouring trees. This phenomenon was
the main reason, according to
, for the observed
non-linearity of the self-thinning line for various tree species.
Observations in virgin forests show that, at the end of a life
cycle, a successive growing space release of the main stand
. Thus, if the ontogenic cycle develops
further, the main stand will collapse at any time. Yield research
generally avoids this stage, in that, as soon as such a collapse
occurs, and an opening or several openings appear, the record
is interrupted. Thus long-term yield plot series represent only
full closed stands. In reality, there is a continuous transition
from full closure to light opening. In any case, an acceptable
working hypothesis seems to be that, during the ontogenic
development of the same tree species, the density relationships
Another question is whether the scale-density relationship
for a particular species is dependent on the site index.
Generalisation of Reineke’s stand density rule implies more or less
independency. Studies of Pinus radiata plantations
and other crops
provide evidence and suggest
that the validity of this assertion should be doubted and,
instead, that harsh site conditions influence competitiveness
differently from good conditions.
When determining maximal stocking, another difficulty
arises. The observed maximal basal area Gmax varies within
a large range, presenting sometimes extreme values beyond
what is usually expected. This is particularly the case in
countries with very different climatic conditions like in the
perialpine zone, especially when precipitation attains a high level.
For spruce stands, for instance, extreme maximal basal area
value of more than 100 m2/ha has been recorded
and Zingg, 2001)
. The phenomenon that density and yield
characteristics can vary substantially independently of the site
index, was first mentioned 1955 by Assmann (1955) under
the term “yield level”. It was first defined as the variation in
the total yield over hg (mean quadratic height), but the yield
level also affects the maximal basal area Gmax as well as the
stem number density and other yield features (Assmann, 1974;
Franz, 1965; 1967) . Basal area can be considered as a good
indicator of the carrying capacity of the site.
describes a different way to assess the yield level with a set of
stand dendrometric variables, or alternatively as well with the
N:dg relationship. The latter is promising because it can be
easily related to Reineke’s way of assessing stand density. In
fact, a scrutiny of a scatter plot of N:dg values (Fig. 1)
reveals a variation in both sides of the mean course. Reineke
uses as a reference for the maximal stand density the outer
line, tangential to the values of the scatter-plot. Albeit
frequently used in this way
(Zhang et al., 2005)
questionable from a statistical as well as from a biological point
of view, because it is based on the assumption that only
extremes values represent the real maximum stocking. In reality,
every value represents full closure. The variation in the stand
density can be more than trivial. For beech and spruce stands
under Swiss conditions it amounts to about 80% (±40% of the
mean value) (
). Other authors demonstrated
similar significant variation
(Assmann, 1974; Bégin and Schütz,
1994; Franz, 1965; 1967; Kennel, 1973; Pfadenauer, 1975)
depending on site conditions not yet entirely explained until
now. Reinecke way of assessing density leads to overestimate
the real density conditions by half of the variation around the
mean. Thus the self-thinning curve should correspond to the
statistical mean course of ln(N):ln(dg). In this case, it may be
interesting to consider the residual variation, i.e. the quotient
Nobserved to Nmean, as the yield level.
Stand density relationships are strictly considered valid
only when derived from untouched stands. In temperate
central Europe, with a tradition of repeated thinning untouched
stands do not represent normality and untouched yield plots
are rare. Overcrowding tends to increase their instability and,
especially for species vulnerable to storms like spruce (
et al., 2006
), threatens their survival. Nevertheless,
allometric relationship during ontogenic development of the stands
could also be verified in treated forests stands under the
condition that the thinning regime is uniform. On the other hand,
a thinned stand provides the advantage that thinning avoids
mortality from overcrowding and ensures more stability in the
upper storey of the stand. Since 1882 (i.e. since foundation
of the IUFRO), West European yield research has harmonized
the definition of thinning especially for the implementation of
yield trial series (Verein Deutscher Forstlicher
Versuchsanstalten, 1902). Thus, especially in Germany and Switzerland, long
yield time series have been recorded under the same
thinning regime, B (moderate thinning from below) up to today,
which fulfils the above mentioned conditions of uniformity of
the thinning regime. B-plots are functionally not very different
from plots in untouched stands as the thinning does not
interfere significantly with the main stand development. In
Switzerland since the 1930s, when Schädelin (Schädelin, 1934)
proposed selective thinning (H according to IUFRO definition),
this thinning regime has been applied throughout the
country so that we also have long series for this regime, and they
should also be scrutinised.
The determination of the yield level in repeatedly thinned
stands with different regimes is more difficult because of
compensative forces between the removed trees and the canopy
closure due to the lateral expansion of the crown. In practice,
the silvicultural history of the stands is normally not known,
at least not in detail. The basal area contains intrinsically the
yield level and appears to be the right indicator for its
determination. The problem is that basal area also depends on
the thinning history.
showed, on the basis
of thinning trials with very different thinning intensities, that
full compensation occurs within some limits of closure
opening, from full closure to a thinning intensity threshold (rule
of the relative mean basal area). Only below the threshold of
the so called critical basal area Gcrit does a clear reduction in
the yield features (increment) occur. In finding a way to assess
the influence on the basal area G of the reduced stand density
i.e. by thinning it appears possible to separate the two
influences mentioned above and as a result to determine correctly
the yield level, for all plots treated and not treated. This
allows enlarging the data base and enhances performing statistic
The aim of the present work is to verify and quantify the
size-density relationship so to be able to predict the
maximal basal area Gmax including its variation (yield level). The
variables included should be as simple as possible so that
the method can be used for practical purposes. The study
was part of a project to construct a modern, distance
independent, growth simulator for even-aged pure spruce (Picea
abies Karst.) and beech (Fagus sylvatica L.) stands in
Switzerland (Model SiWaWA), based on one of the largest data set
from classical yield plots available as well as on several
onetime surveys. Because there are not sufficient complete time
series of untouched stands, the data base was enlarged with
a set of incomplete series and some temporary plots. These
were carefully selected for representativeness according to site
variation and stand closure. Some plots from primeval forest
from Switzerland and for spruce from Slovakia have also been
added in order to cover the mature development stages that are
normally missing in yield series, as long as they correspond
to yield plots regarding tree species composition (pure stands)
and stand closure conditions (only stands with full closure).
2. MATERIALS AND METHODS
The data was taken mainly from permanent yield plots repeatedly
measured over a long time span (1882–2006), mostly from the yield
data base WSL (Swiss Federal Institute for Forest Snow and
Landscape Research, Birmensdorf). The permanent yield plots WSL are
distributed over the whole of Switzerland. Plots above 1 200 m a.s.l.
have been excluded. Plot size generally amounts to about 0.25 ha.
All trees are identified by a number. Measurements were carried out
about every 4 to 7 y after a thinning intervention. The dbh of all trees
above 4 cm dbh (in some case for young stands above 2 cm) is
accurately recorded to mm, crosswise with a calliper. Inventories from the
plots of the chair of silviculture ETH-Z have been added to enlarge
the information base on untreated stands and also to keep records
of the actual kinds of treatment of selective thinning according to
particularly in the old and mature stages. ETH plots
are of two kinds: (1) repeated surveys, similar to the WSL plots in
design and measurement, regarding one thinning trial with three
intensities of selection thinning in spruce (regime H) and four permanents
plots in a pure beech pristine forest reserve St-Ursanne (Tariche) in
Switzerland; (2) selected stands, surveyed once, from the training
forest ETH, typical for the treatment H but with full canopy closure,
as well as 51 plots of untreated stands, selected for a Ph.D.-thesis
. To obtain more information on older development
stages for untreated spruce, which is very important for verifying the
extremities in the N:dg thinning curve, three inventories of a
permanent plot from the virgin forest Kosodrevina (Chopok) Low Tatras,
Slovakia (elevation 1 230) have been included, selected from a
development stage with full closure
(see diameter distribution of the plot
3 in Korpel, 1995, p. 252)
. This is the only comparable spruce
virgin forest available. Other spruce Slovakian virgin forests lie in the
near subalpine belt and could not be considered according to
, because they showed differences between mountain and
subalpine forest dynamics at an altitude threshold of about 1 500 m
Thinning interventions in WSL plots correspond, in terms of
thinning types and intensity, generally to conventional practice, i.e. with
moderate thinning from below until about 1940 i.e. grade B
according to IUFRO rules
(Verein Deutsche Forstlicher Versuchsanstalten,
and selective thinning according to
1940. In some cases yield series with variation in thinning intensity
(A = natural thinning, B = moderate thinning from below, C = heavy
thinning form below, D = very heavy thinning from below, H =
thinning from above) have been established.
The complete data set available consists of 418 surveys for spruce
and 620 for beech. Table I presents the distribution between complete
long times series (30% of the surveys for spruce and 45% for beech),
incomplete long time series (29, 29%), short time series with 2 to
5 surveys (21, 23%) and one time surveys (19, 3%). Table II presents
the age class distribution of the surveys, showing a lack of data for
unthinned older spruce stands, and, but less evident, on unthinned
beech stands. This explains the necessity to incorporate data from
pristine forests. The variation in terms of site index defined as the top
height at age 50 y is 15.5–30.7 m for spruce and 12–26 m for beech,
and in terms of stand density (SDI) 0.403–1.000.
The heights of a sample of 20 to 40 trees at each site were
measured with a hypsometer type Jal with a 7 m reference rod and an
accuracy of about 0.8 m. The site-index, defined as the top height (mean
height of the 100 largest trees per ha) at the age of 50 y (Hdom50) is
determined with the height:age function of the growth simulator
Different models have been tested to explain N (stem numbers)
as dependent from the mean quadratic diameter dg and its square to
assess curvilinearity in model (1) as well as the addition of the
covariables Site Index (SI) (model 2) and the provisional stand density
(PDI) model (3). PDI is the quotient between Nobserved and Nfunction(1)
in the model and corresponds to the factor SDI (site density index) in
Reineke’s terminology sensu. In the latter case, the density is related
to the outer line tangent of the value scatter-plot. In the former case
it is related to the mean curve position. This explains the need for a
ln(Nmax) = α + β × ln(dg) + γ × ln(dg) + ε
ln(Nmax) = α + β × ln(dg) + γ × ln(dg) + δ × S I + ε
ln(Nmax) = α + β × ln(dg) + γ × ln(dg) + δ × S I + ϕ × PDI + ε. (3)
Regression analysis was processed with the statistics package
SYSTAT for windows (Systat software Inc.). The distribution of the
residues was controlled visually and found to be appropriate in every
Figure 1 shows the classical size-density relationship stem
numbers (N) over dg, according to
unthinned stands in a double logarithmic grid for spruce and
beech. The mean curve is fitted with multiple regression
analysis. In order to test the curvilinearity, a quadratic term is
, model (1).
The results of the regression analysis are expressed in terms
of the t-statistic of the standard partial regression coefficient
tolerance of the variables, representative of their usefulness in
the model and the corresponding p-tail of the
transgressionprobability (Tab. III). Significant parameters are highlighted
in bold. A statistically significant parameter γ means that the
curvilinearity of the model applies.
This means that the inclusion of the quadratic component is
Size-density for unthinned spruce and beech
Figures 2a–2c shows the size-density relationship for
different thinning regimes H (selective thinning from above), B
(moderate thinning from below), C (heavy thinning from
below) and D (very heavy thinning from below) for beech stands.
Figure 3 depicts the same for spruce stands, but only
displaying single values for H-thinning. The corresponding t-statistic
for the standard coefficient tolerance for the different variables
is given in Table IV.
In order to verify a possible influence of the site index (SI)
on the size density relationship, a multiple regression is fitted
with the site index (SI) as an additional dependent variable
(Eq. (2)) for the unthinned stands. The site index is defined as
the top height (hdom) of trees aged 50 y. The corresponding
tstatistic for the standard coefficient tolerance of the variables
is given in Table V.
In both cases the influence of the site index appears not to
be significant with a p-tail of 34% and 20%. Because of the
relatively small number of different values in the data set for
unthinned stands, an eventual influence of the site index is
verified with all plots, including the treated stands by
incorporating the stand density and the variable PDI (provisional density
index) into the regression (model 3).
The t-statistic of the standard coefficient tolerance in
model (3) shows that the parameter δ caracterising the
effectiveness of the site index is not significant, with t values for
spruce: t = 1.23 ( p = 0.2179 n.s.) N = 315; R2 = 0.998 and
for beech t = 0.77 ( p = 0.4401 n.s.) N = 661; R2 = 0.998.
Otherwise the parameters γ and φ are highly significant. This
corroborates evidently the results of model (2). It is worth
mentioning in this model (3) that N is not equal Nmax.
Separation of the influence of yield level and stand closure for treated stands
In the case of unthinned stands the basal area is therefore a
very adequate indicator for the stand density. It is functionally
directly dependent on N and dg, intrinsically as a product of N
and dg square. However, density is not identical with canopy
closure, as full closure can occur at very different basal areas,
depending on the yield level. For instance, for adult spruce
stands, full canopy closure is reached with basal area varying
from about 50 m2 up to 74 m2 where the stand characteristics
are the same (site index and age) due to differences in the yield
level, in this case from 0.8 to 1.2, which is absolutely realistic
In thinned plots with unknown thinning history, the
determination of the yield level is more difficult because thinning
affects the basal area. A realistic determination of the yield
level can be assessed if we can determine the portion of basal
area G due to the thinning regime (ΔGth). This has been
performed on data of plots with a known thinning regime. An
estimation of the effect of the thinning regime on the basal area G
has been assessed with the difference between the mean basal
area of all plots (unthinned and thinned) to the mean basal
area G corresponding to the thinning regime under
examination. Thus ΔGth + Gobserved = putative Gmax. Figures 5a and 5b
shows the ΔGth values over the density expressed with PDI for
spruce and beech.
This relationship can be fitted with the following model.
ΔGth = α + β × PDI−2 + γ × (20 − dg)2 + ε (4a for spruce)
ΔGth = α + β × PDI−1.2 + γ × (40 − dg)2 + ε. (4b for beech)
The power coefficient has been determined iteratively, and
the culmination of a quadratic influence of dg visually. The
corresponding t-statistic for the standard coefficient tolerance
for the different variables is given in Table VI.
Thus, for each plot, regardless of its thinning history, the
yield level can be estimated as following:
E N = (ΔG + Gobserved)/Gmean untreated.
Beech, N : dg for treated stands, thinning regime B
Beech, N : dg for treated stands, thinning regime C
Beech, N: dg for treated stands, thinning regime D
N/ha Size-density for different thinning regime (spruce)
Course of maximal basal area
It is not our intention to discuss the generality of the
socalled 2/3 power rule of
Yoda et al. (1963)
applicability to every plant crop, as this has been critically discussed
numerous times. However, our results seem to support
criticism of the universality of this rule
(see Pretzsch, 2000; 2005)
Beech YL 1.0
Linear fit (spruce)
Beech YL 0.8
Spruce YL 0.8
Spruce YL 1.0
Beech YL 1.2
Spruce YL 1.2
Our data material applies to the classical yield of two
important tree species in temperate European forestry. Under
discussion here is the particular applicability of Reineke’s
, determined as the ln(N) to ln(dg)
relationship. It is usually accepted that this relationship is
invariant during the ontogenic development, but it may also
The curvilinearity of the relationship ln(N) to ln(dg) turns
out to apply, at least to the data and statistics recorder in
Switzerland. Thus it seems justified to use equation (1) as the
reference to determine the normal stand density of a growth
simulator. The data base appears to be sufficient and
representative enough for this. Particularly convincing is that the
convex course of the relationship and the form of the
deflection which is comparable throughout the different thinning
regimes and for the two species. These results contradict
previous findings that Reineke’s thinning curve is linear and
(Enquist and Niklas, 2001; Lonsdale, 1990; Weller, 1987;
Zeide, 1987; 2005)
. Whether a change in the steepness of the
relationship during the ontogenic development can be
interpreted as a change in allometry as suggested by
is herewith not definitively verified, because we cannot
exclude that canopy gaps, even if they are very small and
dispersed, had occurred anytime, but they suggest that changes
in closure behaviour occurs. Our data from permanent yield
plots represent principally full closure conditions, as stands
were excluded if the canopy is interrupted. The same applies
to the one-time plots, selected after carefully verification of
their representativeness and their full closure, and especially
to those from unthinned stands after verification of their
There are two methodological sources of bias to be
discussed. Firstly, for very young stands, dg corresponds to the
full, and not to the truncated distribution of the trees due to an
inventory threshold. Our data were verified on the basis of
diameter distributions and, if necessary (in the case of truncated
distributions), excluded from the calculations. Thus N should
consider the entire cohort, but without second growth, under
storey or brushwood. Nevertheless, the convexity of the
thinning curve appears particularly clear for the smaller diameters,
maybe because mortality due to overcrowding manifests itself
only after a certain period of tightness. This would be the case
for a stand with planting with large spacing but does not apply
to our data that developed from dense plantations or natural
regeneration. It is possible that there is a certain delay in the self
thinning process or that very young trees are more resilient to
studied the yield level of spruce in
Bavaria and claimed that young stands (under 40 y) should not
be considered in calculating yield level because it had not yet
reached an equilibrium. In our case, this is not that relevant for
modelling the stand density because transformation of N and
dg into basal area in the case of very young stands does not
affect substantially the level of basal area G (see Fig. 4).
A second methodological source of bias is related to the
logarithmic transformation method regularly used to display
the self thinning course and to calculate the corresponding
regression coefficients. Logarithmic transformation introduces
an important weighting of the values favouring the small over
the larger dimensions. As a matter of principle, this can be
taken into account by weighting the values inversely
proportional to the logarithm. We did not weight our data that it can
be compared with literature. This kind of bias is more
important. Values for large dg are more often underrepresented
because the observations are interrupted following stand
decay or even intentional regeneration. Maintaining full closure
should not be considered as static as it usually is, but a
dynamic process leading normally to progressive canopy
opening. That was a reason for us to verify thoroughly the full
closure conditions, and to incorporate data from primeval (virgin)
forest research plots displaying full closure.
Pretzsch (2005b) considers it necessary to check whether
the thinning curve is linear, case by case on long time series
(chronosequences) for the same stand. He found that 2 out
of 9 long time series were curvilinear for spruce and 3 out
of 9 for beech. We agree principally with this argument,
although the problem of the definition of real closure conditions
and successive canopy opening remains the same as for
scatter plots. We have only a few time series at our disposal and
only a few repeated observations (mean 6.4 spruce and 8.4
for beech). For spruce particularly very old values are
lacking. Therefore, Pretzsch’s approach is not quite appropriate for
our data. Nevertheless, we calculated for all chronosequences
with more than 5 inventories the regression according to
equation (1) including A (natural thinning) and B plots (moderate
thinning from below). At a level of transgression probability
of 5% it appears that 1/3 for spruce A-plots, 1/10 of spruce
B-plots, 2/3 of beech A-plots and 6/14 of beech B-plots have
significantly negative γ. No positive γ was found, in contrast
Figure 4 also shows the substantial consequences of using
a curvilinear fitting of the N:dg curve in comparison to the
classical linear fitting. In terms of basal area, the difference is
impressive, especially over dg 40.
In conclusion, in some cases there is evidence of a convex
curvilinear thinning curve, especially in beech. When
considering a large distribution of plots, it appears as a clear
tendency. Because mortality due to hazards is largely, although
not absolutely, excluded from our material the phenomenon
could be due to changes in canopy utilization. A possible
explanation is that mortality due to overcrowding, affecting
social low elements in young stands, changes with time, affecting
more elements of the main canopy in adult stages and
especially ageing stages. Pretz
; 2008) presents results for
spruce and suggests as a factor retraction of the crown due to
swaying movements as well as the geometry of space
occupancy and other factors.
Differences between the spruce and beech stands were
substantial, paticularly the level of basal area G. The stocking of
full closed beech stands culminate at approx. 45 m2 in
comparison with approx. 65 m2 for spruce (see Fig. 4), i.e. about 50%
more for spruce. This corroborates the differences in growth
efficiency found by
Pretzsch and Schütze (2005)
who found a
better space occupancy of spruce and a better space
exploitation of beech due to differences in branching and resource
allocation. The lightly more convex form of the G-course for
beech corresponds to a greater change in N:dg steepness of the
thinning curve, as well as the more lateral expansion of the
One of the valuable results is the independence of the
sizedensity and the site index. A first visual analysis of the
distribution of N:dg values from different site index classes (not
shown here) revealed no pattern supporting such an effect. The
statistical analysis through regressions analysis confirms this.
The range of site index varies between 15.5 and 30.7 m hdom
for trees aged 50 for spruce and between 12.0 and 26.0 m for
beech which is rather a wide range. It seems that height
differences as encapsulated in the site index explain the space
occupancy behaviour sufficiently.
One of the interesting results is that the classical way of
assessing the productivity of a site with the site index based on
the height:age relationship does not influence particularly the
size density relationships. On the other hand, the basal area G
varies substantially, give reasons to reconsider the validity of
the site index utilization. For regions with a great site variation
(climate, precipitation, soil, aspect), where substantial
variation of yield level could be expected, the basal area G is much
more appropriate to characterize the yield performances than
age to height. This is illustrated in Figure 4 where the range of
variation in basal area G is much wider between the low and
the high yield level than variations in stand development (dg or
age). This does not mean that the site index, which gives the
third dimension (stand height), is not useful. To determine all
three dimensional variables like volume, the site index is
determinant. Thus we need to determine two components of the
yield capacity: its horizontal dimension expressed by the yield
level and its vertical by the site index. In our study the
former revealed more influential than the latter. The growth
simulator SiWaWa, for instance, is elaborated on the assumption
that basal area G and mean quadratic diameter are sufficient
as primary entry variables in the model. It is not necessary
to determine the site index with high precision. Its estimation
with floristic maps or broad geographic and climatic variables
. Even age is not considered as a
primary variable, as the stand development can be easily
characterized by dg, with age derived indirectly. This simplifies
the practical application because age is often not known and
height is difficult to assess. This approach complies with in
the spirit of Langton’s think movement/theory of so called
, which aims to explain a complex
system with criteria that are as simple as possible.
Up to now the yield tables and other growth models
represent only full closed stands or regular opening of the canopy
in the case of thinning regimes. In reality, as soon as canopy
interruptions occur, such models are no longer reliable. Thus
yield models are generally not appropriate for projecting stand
development, or only under certain conditions. The difficulty
is that such mortality events from disturbances are stochastic
(Drössler and Lüpke, 2005; Quine and Bell, 1998; Schütz et al.
2006; Zeibig et al., 2005)
and assessing the risk occurrence
implies different methodological probabilistic from those used
to estimate yield alone. For instance, it needs largely
corresponding areas covering hundreds of hectares, way beyond the
usual reference for yield plots. Furthermore, the risk of canopy
dislocation varies with regional frequency of storms and with
advancement of ageing factors. From research on comparable
virgin forest, we know that the life cycle duration seems to
present some constancy
, 250 y for beech
between 300 and 350 for spruce. Within this time the stand will
have been completely renewed. Nevertheless the inclusion of
risk in growth models is an interesting prospect that requires
Yield level has been interpreted as a potential for a
horizontal space occupancy
, as variable properties for
competition (and for self-thinning) so that a stand can afford
more (or fewer) stem numbers (
). Pretzsch (2000)
considers it as an expression of the site carrying capacity. A
high yield level leads to intensified tightness (expressed as
stem density, see Fig. 6), and thus to reductions in the
individual increment. But the proportionally higher number of trees
per area tends to enhance slightly the total increment (
). The variations in the yield level have not yet been
completely explained. It seems realistic to consider that they could
have to do with the limitations of the site resources (water,
nutrients). Some general dependencies with broad sites factors,
such as precipitation and elevation, are discernable (
), but these do not explain the whole variation. Other
attempts to include edaphical orographical variables are
interesting, but there is methodological problem with assessing
accurately and representatively the site characteristics, particularly
for edaphic variables
(Bégin and Schütz, 1994; Schütz and
. Wide variations in yield levels within short
distances have been observed (Pretzsch, personal
communication), even in the same communal location (Schütz, 1992).
The fact that the yield level can be assessed correctly with the
basal area G is interesting and applicable for modelling
purposes, but more research is needed to explain the relationship.
Acknowledgements: We thank colleague Milan Saniga, Professor
of Silviculture at the Technical University Zvolen, Slovakia for
providing data form the virgin forest plots on Slovakia and Silvia
Dingwall for bringing the text closer to good English. We are also grateful
to two anonymous reviewers for helpful comments on an earlier draft
of this manuscript.
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