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ŠUMARSKI LIST 11-12/2009 str. 29     <-- 29 -->        PDF

S.Andrašev, M. Bobinac, S. Orlović: DIAMETER STRUCTURE MODELS OF BLACK POPLAR SELECTED ... Šumarski list br. 11–12, CXXXIII (2009), 589-603

response to environmental conditions
(primarily soil conditions and
tree competition) of each genotype.

The sample of trees used for the
construction of the diameter structure
model (total 168 samples) comprised
the measured diameters at
breast height (mean value of two
cross measurements) in each row
(4 repetition),for each clone (6 clones)
and measurement year (7 years).
Basal area, as the sum of basal areas
of all trees, and the stand quadratic
mean diameter were calculated for
each tree samples.

Table 1
Number of trees of studied clones by repetitions after planting and after
20 years.

Tablica 1.Broj stabala istraživanih klonova po ponavljanjima pri osnivanju
nasada i nakon 20 godina.

Number of trees after
planting per repetitions
Broj stabala nakon sadnje
po ponavljanjima
Number of trees after
20 years per repetitions
Broj stabala nakon 20
godina po ponavljanjima
25 25 25 25 23 24 24 24
22 22 22 22 21 20 21 20
25 25 25 25 23 23 24 23
Pannonia 25 25 25 25 22 24 22 24
PE19/66 20 20 20 20 19 17 19 19
25 25 25 25 24 23 22 23

The selected model was the Weibull distribution
with three parameters.The mathematical model of the
Weibull distribution is defined as follows:

where:a– location parameter;b– scale parameter;
c – shape parameter. The mathematical model of the
Weibull cumulative distribution is expressed as:


Location parameter (a) defines the location distribution
in the coordinate system, i.e. its distribution
along the abscissa. Scale parameter (b) is equal to 63%
of the distribution of unknown value (x) in the increasing
order, i.e. about 63%of the trees have diameter at
breast height lower than the sum of parameters “a”and
“b”. Shape parameter (c) defines the distribution skewness:
forc<1 the distribution decreases, and forc>1 it
has bell shape. In the interval 1is positively skewed, forc>3.6 it is negatively skewed,
and for c=3.6 it approximates the normal probability
density function (PDF).

If the location parameter (a) is equal to zero the
mathematical model turns into the so-called two-parameter
model of theWeibull distribution, defined by the

The parameters of the Weibull probability density
function can be estimated in several ways (from sample
trees), depending on the desired estimation of two (b,
c) or all three parameters (a,b,c).The unidentified parameters
of the Weibull distribution were estimated
using the so-called “hybrid system”, i.e. the method of
moments estimation in combination with the percentile
method (Knoebel, et al, 1986). Location parameter

(a)in diameter structure modelling is directly related to
minimal diameter and can vary from 0 (zero) todmin. So
the parameter “a” was calculated by the percentile
method, with the following percentiles of the minimal
diameter: 0.00; 0.01; 0.05; 0.10; 0.15; 0.20; 0.25; 0.30;
0.35; 0.40; 0.45; 0.50; 0.55; 0.60; 0.65; 0.70; 0.75;
0.80; 0.85; 0.90; 0.95; 0.99; 1.00.

Scale parameter (b) and shape parameter (c) were
estimated by the moments method.They were estimated
by subtracting the measured values (diameters at
breast height) from the previously defined parameter
“a”.This method is based on the following equations



of the first (x) and the second (x) common moment of
theWeibull two-parameter distribution:



where.(·) – gamma function.
The assessed variance () of theWeibull distribution
is expressed as:

and the coefficient of variation ():

and the cumulative model: