DIGITALNA ARHIVA ŠUMARSKOG LISTA
prilagođeno pretraživanje po punom tekstu




ŠUMARSKI LIST 11-12/2009 str. 27     <-- 27 -->        PDF

IZVORNI I ZNANSTVENI ČLANCI – ORIGINAL SCIENTIFIC PAPERS Šumarski list br. 11–12, CXXXIII (2009), 589-603


UDK 630* 165 + 561 (001)


DIAMETER STRUCTURE MODELS OF BLACK POPLAR SELECTED
CLONES IN THE SECTION AIGEIROS
(DUBY) OBTAINED
BYTHE WEIBULLDISTRIBUTION


MODELI DEBLJINSKE STRUKTURE SELEKCIONIRANIH KLONOVA
CRNE TOPOLE SEKCIJE AIGEIROS (DUBY) DOBIVENI
WEIBULL-OVOM DISTRIBUCIJOM


1 23


Siniša ANDRAŠEV, Martin BOBINAC, Saša ORLOVIĆ


ABSTRACT: The present study was performed in an experimental plantation
with six 20-year-old black poplar clones in the Section Aigeiros(Duby). The
diameter structure models were constructed using the Weibull probability density
distribution with three parameters based on periodical measurements of
diameters at breast height. The unidentified parameters were calculated by the
so-called “hybrid system” (Knoebel, et al, 1986): location parameter (a) was
calculated by percentile method, scale parameter (b) and shape parameter (c)
were calculated by moments method. The applied method of estimating the location
parameter (a) showed that in 90.6 % of the study sample, the parameter
“a” ranged between 50 and 90 %, and in 52.4 % of the sample, “a” ranged
from 80 to 90 % of the minimal diameter. With higher plantation ages, location
parameter (a) and scale parameter (b) also increased with small oscillations,
which was confirmed by the significance of the correlation coefficient of 0.71
and 0.73 respectively. This was shown by the shift of the curve of diameter
structure model to the right, towards larger diameters, and in a wider range of
diameters at breast height with a lower relative frequency of the modal degree.
In the initial period, F-ratio of all three parameters of diameter structure model
decreased and reached the minimal value in the eighth year, and the predominantly
increasing trend started in the twelfth year, which points to the changes in
diameter structure of the study clones depending on the age. The plantation
growth elements (dg, G) and the Kolmogorov-Smirnov test, as well as the analysis
of variance test and LSD test for the percentage of the number of trees with
diameters at breast height above 40 cm, confirmed the grouping of diameter
structure models of the study poplar clones in two groups. This makes it possible
to define the differentiated management procedures with individual groups.


Key words:black poplar, clones, diameter structure, Weibull distribution.


1. INTRODUCTION – Uvod


The main parameters which characterise poplar plan-bioecological and development-production characteritation
production are: poplar clone (cultivar) and its stics, the site with its specificities, and the technologies
of plantation establishment, tending and protection, inc


1


Dr. Siniša Andrašev, research associate, Institute of Lowland
Forestry and Environment,Antona Čehova 13d, 21000 Novi Sad,


luding also plantation density depending on the specific
Serbia; E-mail: andrasev@uns.ac.rs


purpose.All the above parameters are interdependent,


2


Dr. Martin Bobinac, assistant professor, University of Belgrade


but the correct selection of poplar cultivar is of primary


– Faculty of Forestry, Kneza Višeslava 1, 11030 Belgrade,


importance for the optimal use of site potential.


Serbia; E-mail: mbobinac@EUnet.rs


3


Dr. Saša Orlović, principal research fellow, Institute of Lowland


Compared to natural ecosystems, poplar plantations


Forestry and Environment,Antona Čehova 13d, 21000 Novi Sad,


of selected new cultivars and clones ensure multiple


Serbia; E-mail: sasao@uns.ac.rs




ŠUMARSKI LIST 11-12/2009 str. 28     <-- 28 -->        PDF

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


increases in the dendromass quantity and value. By the
establishment of intensive plantations after the Second
World War in Serbia, in a relatively short period (20–25
years), annual poplar and willow felling volume increased
12 times, and by improving the assortment
structure, the value of wood volume increased almost
20 times compared to the pre-war period (Marković,
et al., 1995). However, plantation establishment with
one clone over large areas soon resulted in susceptibility
to pathogen and pest attacks. For this reason, one of
the strategic directions of further poplar research was
the continuing creation (selection) of new poplar clones
and their putting to production. These processes
were accompanied by the quantification of productivity
at the designated sites and for the selected plantations
management procedures.


The quantification of differences between genotypes
is usually defined by mean values of growth elements:
diameter (d), height (h), and the derived elements, basal
area (G), and volume (V).Also, the study of internal
stand (plantation) structure makes it possible to analyse
the state and to project the future development which
leads to a more reliable identification of productivity
differences. In this process, modern forest management
planning applies the models of tree and stand (plantation)
growth, based on real and testable data which also
include the data on ecological conditions and growth
characteristics of forest trees.The first step in the design
of stand models is the model of diameter structure.


Diameter structure modelling in even-aged stands
(and plantations) applies theWeibull probability density
function.The mathematical model of the function was
defined by Weibull,(1951) in his study of the reliability
of material hardness. It was introduced to forestry by
Bailey andDell (1973) who constructed the model
of stand diameter structure. Since then, theWeibull distribution
has been widely implemented in forestry because
it can describe a wide range of unimodal
distributions and it can be adapted to both negative and
positive skewness (Bailey and Dell, 1973; ......,
1984;Zarnoch andDell,1985;Knowe, et al.,
1994;Kotar,2005). AccordingtoBailey andDell,
(1973), ...... (1984), a special characteristic of
Weibull distributionis the fact that its parameters have a
biological interpretation. The significance of the Wei-
bull distribution in the construction of diameter structure
model in poplar plantations was reported by Andraševet
al.(2003, 2004),Andrašev(2008).


The objective of this study was to investigate the
suitability of theWeibull distribution for the construction
of diameter structure models of newly selectedpoplar
clones, SectionAigeiros(Duby), by applying the
so-called “hybrid system”of predicting the model parameters
from the sample.Also, the objective was to
study the change in model parameters depending on
plantation age and poplar clone, and also their relation
to plantation growth elements (dg ,G).


2. MATERIAL – Objekt istraživanja


The research was performed in a 20-year-old test
plantation consisting of several clones (cultivars) of
black poplar in the SectionAigeiros(Duby). Theplantation
is located on the experimental field of the Institute
for Lowland Forestry and Environment (former
Poplar Research Institute) near Novi Sad, planting
space 5 × 5 m (400 trees per hectare), plant type 2+0.
The plantation soil is fluvisol, sandy-loamy form (Škorić
et al.,1985) and can be considered as medium suitable
for poplar growing. The following poplar clones
(cultivars) were researched:
1S (Populus deltoidesBartr. ex Marsh.) – cultivar


6-36


registered in Serbia in 1987;
2NS (Populus deltoidesBartr. ex Marsh.) – cultivar


1-3


registered in Serbia in 1998;


3NS (Populus × euramericana (Dode) Guinier)


11-8


× (Populus deltoidesBartr. ex Marsh.) – cultivar registered
in Serbia in 1998;
3 Pannonia (Populus × euramericana (Dode) Guinier)
– cultivar registered in Serbia in 1998;
5 PE 19/66 (Populus deltoidesBartr. ex Marsh.) – in
selection procedure;
6S(Populus deltoidesBartr. ex Marsh.) – in selec


6-7


tion procedure.
Each clone in the test plantation consisted of six
rows, with 20–25 trees per row.The fringe rows were
not included in measurement and processing, because
of the mutual influences. From the aspect of the experiment,
each row represents a replicate (altogether 4 replicates)
for the statistical processing of the results.


3. METHOD – Metoda rada


Diameters at breast height of all trees were periodically
measured (to the nearest 1mm) after one, two, five,
eight, twelve, seventeen and twenty years from the test
plantation establishment. The number of trees in the
plantation decreased over time due to different causes:
after 20 years, minimum 85%of the initial num ber of
trees remained in each row (replicate) (Table 1). Their
diameters measured at the above ages were used for the
construction of the diameter structure model.This was
done to avoid the impact of changes in parameters due to
the decrease in the number of trees.As the same trees
were measured throughout the study period, the parameter
changes in diameter structure models can mostly be
assigned to the process of tree growth, i.e. to the specific




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.


Clone
Klon
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
I II III IV I II III IV
S
6-36
25 25 25 25 23 24 24 24
NS
1.3
22 22 22 22 21 20 21 20
NS
11-8
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
S
6-7
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:


(1)
where:a– location parameter;b– scale parameter;
c – shape parameter. The mathematical model of the
Weibull cumulative distribution is expressed as:



(2)


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
expression:


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





-


2


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


(5)


(6)


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


(3)
(7)
and the coefficient of variation ():


and the cumulative model:


(8)


(4)


ŠUMARSKI LIST 11-12/2009 str. 30     <-- 30 -->        PDF

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


By calculating the first and the second common mo-The percentiles of minimal diameter were selected


2


ment, and also the coefficient of variation from the
sample and by inserting in formula 8, the coefficient of
variation is the function of only one parameter (c), so it
can be estimated by an iteration procedure.The combination
of secant method and bisection method was ap


-6


plied with the previously set precision of 10 (Conte
and deBoor,1980).
The value of parameter “c” was applied to obtain
the value of parameter “b”in the relation:


(9)


Parameter “a”was calculated for each of the above
percentiles in each replicate and the parameters “b”and
“c”by the moments method.Then the empirical diameter
structure and the model were compared usingAnder


2


son-Darling statistic (A) (Anderson andDar ling,
1954):


(10)


where:F(x)– cumulative model of diameter structure;
n– number of samples.


based on the minimal value ofA statistic in all 4 replicates
within the same clone and plantation age.


For each of the selected percentiles of minimal diameter
within each clone and plantation age, all three
parameters of theWeibull distribution were re-estimated
for each replicate. The diameter structure model
and the empirical distribution were tested by non-parametric
Kolmogorov-Smirnov test, using |D| statistics:


(11)


where:F1(x)– cumulative model of diameter structure;
F2(x)– empirical cumulative diameter structure in
the increasing order.


The obtained parameters of theWeibull distribution
per individual replicates were applied in the assessment
of differences between the study clones at certain ages
(years after planting), using the analysis of variance
test and the least significant difference test (LSD), at
the 5%risk level.


Finding the value of the unknown parameters of the
model ofWeibull diameter distribution, by the above
method, was performed programming inVisual Basic,
which is an integral part of the Excel package. For a
statistical assessment STATISTICA, ver. 7.0 software
package was used.


4. RESULTS – Rezultati istraživanja


4.1. Growth elements of the study clone plantations – Elementi rasta nasada istraživanih klonova
The clones attained close stand quadratic mean dia-quence of the uniform dimensions of the applied plan-


meters (dg) only in the first and the second years after ting material, and also of the low increment, especially
planting (Table 2, Diagram 1a), which is the conse-in the first year after planting (Andrašev etal. 2006).


Table 2 Stand quadratic mean diameter (d) and the results of the analysis of variance test and LSD test at the risk level of


g


0.05for the clones per years of measurement.


Tablica 2.Srednji promjer po temeljnici (dg) i rezultati testa analize varijance i testa NZR na razini rizika od 0,05 istraživanih
klonova po godinama izmjere


Clone
Klon
Plantation age after planting -Starost nasada nakon sadnje
1 year
1. god.
2 years
2. god.
5 years
5. god.
8 years
8. god.
12 years
12.god.
17 years
17. god.
20 years
20. god.
dg [cm]
S
6-36
2.9 (0.06)
1
ab
2
6.5 (0.16)ab 17.2 (0.57)c 23.4 (0.48)cd 28.1 (0.39)b 33.1 (0.88)bc 35.0 (1.03)bc
NS
1.3
3.3 (0.08)a 6.5 (0.14)ab 18.7 (0.79)ab 24.9 (1.11)b 28.5 (1.49)b 32.4 (1.77)bc 34.6 (2.16)bc
NS
11-8
2.8 (0.24)b 6.2 (0.36)b 17.7 (1.09)bc 24.7 (0.90)b 29.4 (1.14)ab 34.4 (1.25)b 36.8 (1.20)b
Pannonia 3.0 (0.16)ab 6.5 (0.22)ab 15.9 (0.50)d 22.8 (0.15)d 28.4 (0.37)b 33.7 (0.89)b 35.6 (1.31)bc
PE19/66 3.3 (0.13)a 6.9 (0.24)a 19.0 (0.56)a 26.4 (0.61)a 31.0 (1.05)a 36.7 (1.37)a 40.3 (1.79)a
S
6-7
3.0 (0.09)ab 6.8 (0.20)ab 18.4 (0.87)ab 24.2 (0.89)bc 28.2 (1.80)b 31.3 (1.83)c 33.7 (2.05)c
F 2.29
ns
1.93
ns
10.73
***
11.93
***
3.68
*
7.65
***
8.06
***


1


Values in parentheses represent the standard deviation –Vrijednosti u zagradi predstavljaju standardnu devijaciju.


2


The same letters indicate that there is no statistically significant differences between the clones tested by least significant


differences at the risk level of 0.05 – Ista slova znače da ne postoje statistički značajne razlike između klonova po testu


najmanje značajne razlike na razini rizika od 0,05.


ns* **


non significant –nije signfikantno; significant at the risk level of 0,05 –signfikantno na razini rizika od 0,05; significant


***


at the risk level of 0,01 –signfikantno na razini rizika od 0,01; significant at the risk level of 0,001 –signfikantno na
razini rizika od 0,001.




ŠUMARSKI LIST 11-12/2009 str. 31     <-- 31 -->        PDF

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


The clones were differentiated by mean diameter in the
fifth year after planting and it lasted to the end of the
study period of 20 years.


The clone differentiation per attained basal areas per
hectare (G), which indicates indirectly the differences in
volume per hectare, also, occurs in the fifth year and
lasts through the eighth year (Table 3, Diagram 1b). At


the age of 12, the values of basal area per hectare were


2 -1


very close for all six study clones (24.2–26.3 m·ha )


with F-ratio of the analysis of variance amounting to
only 0.53.With the increasing plantation age, the differences
between clones in basal areas per hectare incre ase
and at the age of 20, the differences were significant
at the risk level of 0.05.


2 -1


a)Stand quadratic mean diameter d(cm)
b)Basal area per hectare, G (m·ha)


g


Srednji promjer po temeljnici dg(cm)
Temeljnica po hektaru G (m2·ha-1)


Diagram 1 Mean values of the stand quadratic mean diameter (dg) and total basal area per hectare (G) of the clones depending on
plantation age.


Grafikon 1. Srednje vrijednosti prsnog promjera po temeljnici (dg) i ukupne temeljnice po hektaru (G) istraživanih klonova u zavisnosti
od starosti nasada


Table 3
Basal area per hectare (G) and the results of the analysis of variance test and LSD test at the risk level of 0.05
for the clones per years of measurement.


Tablica 3.Temeljnica po hektaru (G) ) i rezultati testa analize varijance i testa NZR na razini rizika od 0,05 istraživanih
klonova po godinama izmjere.


Clone
Klon
Plantation age after planting –Starost nasada nakon sadnje
1 year
1. god.
2 years
2. god.
5 years
5. god.
8 years
8. god.
12 years
12.god.
17 years
17. god.
20 years
20. god.
Gg [m2 ·ha-1]
S6-36 0.27(0.02)
1
bc
2
1.31 (0.11) a 9.10 (0.73) b 16.83(0.55) bc 24.34 (1.01) a 33.67(1.22)abc37.63 (1.42) bc
NS1.3 0.32 (0.03) ab 1.24 (0.11) a 10.47 (0.99) a 18.69(2.32) ab 24.19 (3.17) a 31.56(4.32) bc 36.13 (5.42) bc
NS11-8 0.24 (0.06) b 1.22 (0.17) a 9.35 (1.13) ab 18.12(1.65) ab 25.70 (2.40) a 35.10(2.90) ab 40.31 (2.92) ab
Pannonia 0.27 (0.01) bc 1.33 (0.12) a 7.62 (0.69) c 15.67 (0.80) c 24.32 (1.15) a 34.23(1.62)abc38.29 (2.42) bc
PE19/66 0.34 (0.06) a 1.48 (0.14) a 10.57 (0.12) a 19.12 (1.05) a 26.30 (1.64) a 37.09 (3.08) a 44.70 (4.36) a
S6-7 0.28 (0.01) ab 1.46 (0.05) a 10.34 (1.32) ab17.99 (1.77) ab 24.61 (3.74) a 30.14 (4.09) c 34.32 (4.11) c
F 2.15ns 1.85ns 5.54** 2.94* 0.53ns 2.59ns 3.85*


1


Values in parentheses represent the standard deviation –Vrijednosti u zagradi predstavljaju standardnu devijaciju.


2


The same letters indicate that there is no statistically significant differences between the clones tested by least significant


differences at the risk level of 0.05 – Ista slova znače da ne postoje statistički značajne razlike između klonova po testu


najmanje značajne razlike na razini rizika od 0,05.


ns*
**


non significant –nije signfikantno; significant at the risk level of 0,05 –signfikantno na razini rizika od 0,05; significant


***


at the risk level of 0,01 –signfikantno na razini rizika od 0,01; significant at the risk level of 0,001 –signfikantno na
razini rizika od 0,001.




ŠUMARSKI LIST 11-12/2009 str. 32     <-- 32 -->        PDF

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


Test results of the analysis of variance of repeated
measurements showed a high (G) and very high (dg) significant
interactions clone×year which indicates that
the investigated clones had different increments of the
quadratic mean diameterand total basal area per hectare
in the study period (Table 4, Diagram 1).


Throughout the period of 20 years, the clone PE
19/66 attained the best results both in stand quadratic
mean diameter (dg), and in total basal area per hectare


(G) and it is differentiated from the other clones by
LSD test at the 5%risk level.


Clone Pannonia attained the lowest sizes of the


stand quadratic mean diameter and the total basal area


per hectare in the period to the eighth year. In further
development, its increment was more intensive compared
to other clones and at the age of 20 years, its mean
diameter and total basal area per hectare were greater


compared to the clones S , NS and S .


6-7 1-3 6-36


The growth of clone NS was also more intensive


11-8


with the increased age.


Table 4
Test results of the analysis of variance of repeated measurements ofthe quadratic mean diameter
and total basal area per hectare.


Tablica 4.Rezultati testa analize varijance ponovljenih mjerenja srednjeg promjera po temeljnici i ukupne
temeljnice po hektaru.


Sum of Square Degr. of Freed. Mean Square F p
Stand quadratic mean diameter - d
g
[cm]– Srednji promjer po temeljnici - dg[cm]
Intercept 77431.36 1 77431.36 17478.1 0
Clone 147.31 5 29.46 6.65 0.00114
Error 79.74 18 4.43
Year 23945.81 6 3990.97 8612.37 0
Clone ×Year 120.13 30 4.00 8.64 0
Error 50.05 108 0.46
Basal area per hectare – G [m
2
·ha
-1
]– Temeljnica po hektaru – G [m2·ha-1]
Intercept 50759.86 1 50759.86 1461.36 0
Clone 134.79 5 26.96 0.77614 0.57951
Error 625.22 18 34.73
Year 30971.14 6 5161.86 1267.97 0
Clone ×Year 240.87 30 8.03 1.97228 0.00602
Error 439.66 108 4.07


4.2. Diameter structure models obtained by the weibull distribution


Modeli debljinske strukture dobiveni weibull-ovom distribucijom


4.2.1. Location parameters (a) of the Weibull diameter structure model


Definiranje parametara položaja (a) modela Weibull-ove debljinske strukture


Table 5 presents the percentile minimum diameter son-Darling statistics of studied clones in certain years of
with a minimum value of Anderson-Darling statistics surveying. Diagram 2 shows the percentage of the per


2


(A), and mean values and standard deviations ofAnder-centiles of minimal diameter for the definition of location


2


Table 5
Percentile values of the minimum diameter (%d ) that have the least value ofAnderson-Darling (A) statistic of


min


examined clones by years of survey.


Tablica 5.Vrijednost percentila minimalnog promjera (%dmin) koji imaju najmanju vrijednost Anderson-Darling statistikue
(A2) istraživanih klonova po godinama izmjere.


Clone
Klon
%d
min
A
2 1
Age of the plantation after planting
Starost nasada nakon sadnje
Age of the plantation after planting
Starost nasada nakon sadnje
1. 2. 5. 8. 12. 17. 20. 1. 2. 5. 8. 12. 17. 20.
S
6-36
0.85 0.90 0.85 0.85 0.90 0.90 0.90 0.33(0,15)
2
0.60 (0,06)0.66 (0,21) 0.67 (0,12) 0.49 (0,29) 0.28 (0,10) 0.31 (0,16)
NS
1.3
0.90 0.90 0.85 0.75 0.65 0.65 0.55 0.36 (0,17) 0.26 (0,03)0.71 (0,26) 0.46 (0,23) 0.59 (0,21) 0.37 (0,17) 0.51 (0,39)
NS
11-8
0.60 0.55 0.75 0.80 0.80 0.85 0.85 0.71 (0,25) 0.43 (0,27)0.51 (0,10) 0.46 (0,11) 0.43 (0,19) 0.45 (0,10) 0.38 (0,04)
Pannonia 0.10 0.45 0.70 0.85 0.90 0.80 0.75 0.56 (0,28) 0.50 (0,26)0.43 (0,14) 0.56 (0,27) 0.59 (0,31) 0.20 (0,03) 0.21 (0,03)
PE19/66 0.80 0.70 0.90 0.90 0.90 0.70 0.60 0.23 (0,06) 0.36 (0,04)0.60 (0,21) 0.39 (0,27) 0.36 (0,13) 0.36 (0,09) 0.28 (0,08)
S
6-7
0.45 0,70 0.95 0.80 0.75 0.60 0.60 0.23 (0,06) 0.29 (0,01) 0.55 (0,11) 0.57 (0,29) 0.55 (0,37) 0.55 (0,41) 0.40 (0,24)


1


mean values ofAnderson-Darling statistic– srednje vrijednosti Anderson-Darling statistike.


2


Values in parentheses represent the standard deviation –Vrijednosti u zagradi predstavljaju standardnu devijaciju.
594




ŠUMARSKI LIST 11-12/2009 str. 33     <-- 33 -->        PDF

ns


4.2.2. Comparison of parameters of diameter structure model between clones


Usporedba parametara modela debljinske strukture između klonova


Depending on plantation age, the results of the analy-structure model showed the decreasing trend and the misis
of variance test showed mostly significant differen-nimal value occurred in the eighth year, and the predoces
between the parameters of Weibull diameter minantly increasing trend started in the twelfth year. In
structure model (Tables 6, 7 and 8). In the initial period, the eighth year, there were no significant differences
F-ratio of all three parameters of theWeibull diameter between parameter “c”of the diameter structure mo del,


Table 6
Mean values of location parameter (a) of theWeibull diameter structure model and the results of the analysis
of variance test and LSD test at the risk level of 0.05 of study clones per years of measurement.


Tablica 6.Srednje vrijednosti parametra položaja (a) modela Weibull-ove distribucije prsnih promjera i rezultati testa
analize varijance i testa NZR na razini rizika od 0,05 istraživanih klonova po godinama izmjere.


Clone
Klon
Plantation age after planting -Starost nasada nakon sadnje
1 year
1. god.
2 years
2. god.
5 years
5. god.
8 years
8. god.
12 years
12.god.
17 years
17. god.
20 years
20. god.
S6-36 1,67 (0.09)
1
b
2
4,65 (0.30) a 12,75 (0.69) b 17,42 (0.85) b 22,72 (0.86) a 26,56 (0.98) a 27,98 (1.12) a
NS1.3 2,23 (0.20) a 4,81 (0.43) a 13,18 (1.77) b 15,19 (2.48) c 14,62 (2.88) c 15,71 (2.36) c 13,9 (1.98) e
NS11-8 0,91 (0.18) c 2,54 (0.20) d 10,88 (0.75) c 16,4 (1.39) bc 18,4 (2.69) b 23,18 (3.74) b 24,63 (4.09) b
Pannonia 0,16 (0.01) d 2,10 (0.12) e 9,1 (0.57) d 17 (0.00) bc 22,5 (1.27) a 23,08 (1.31) b 22,29 (1.41) bc
PE19/66 2,09 (0.02) a 4,05 (0.15) b 15,98 (0.45) a 20,92 (0.45) a 24,75 (0.52) a 21,08 (0.76) b 19,59 (0.73) cd
S6-7 0,76 (0.05) c 3,46 (0.36) c 15,2 (1.10) a 16,6 (1.20) bc 17,62 (2.33) b 15,34 (2.17) c 16,41 (2.43) de
F 190.29*** 61.1*** 27.11*** 8.75*** 14.88*** 17.45*** 21.66***


1


Values in parentheses represent the standard deviation– Vrijednosti u zagradi predstavljaju standardnu devijaciju.


2


The same letters indicate that there is no statistically significant differences between the clones tested by least significant


differences at the risk level of 0.05– Ista slova znače da ne postoje statistički značajne razlike između klonova po testu


najmanje značajne razlike na razini rizika od 0,05.


nsnon significant –nije signfikantno; * significant at the risk level of 0,05 –signifikantno na razini rizika od 0,05;
** significant at the risk level of 0,01 –signfikantno na razini rizika od 0,01; *** significant at the risk level of 0,001 –
signifikantno na razini rizika od 0,001.




ŠUMARSKI LIST 11-12/2009 str. 34     <-- 34 -->        PDF

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


which points out the changes in diameter structure of the
study clones depending on the age.


With higher plantation ages, location parameter (a)
and scale parameter (b) also increased with small oscillations,
which was confirmed by the significance of the
correlation coefficient of 0.71 and 0.73 respectively
(Diagram 3).This was shown by the shift of the curve
of diameter structure model to the right, towards larger
diameters, and in a wider range of diameters at breast
height with a lower relative frequency of the modal degree
(Diagram 4).The changes in shape parameter (c)
were smaller, which was confirmed by the correlation
coefficient of 0.36 (Diagram 3). However, its significance
at the risk level of 0.001 indicates the increasing
trend with plantation age, i.e. the change in the shape of
diameter structure positive skewness, or approximately
normal distribution, to the negative skewness.


Table 7
Mean values of scale parameter (b) of theWeibull diameter structure model and the results of the analysis
of variance test and LSD test at the risk level of 0.05 of study clones per years of measurement.


Tablica 7.Srednje vrijednosti parametra skaliranja (b) modela Weibull-ove distribucije prsnih promjera i rezultati testa
analize varijance i testa NZR na razini rizika od 0,05 istraživanih klonova po godinama izmjere.


Clone
Klon
Plantation age after planting –Starost nasada nakon sadnje
1 year
1. god.
2 years
2. god.
5 years
5. god.
8 years
8. god.
12 years
12.god.
17 years
17. god.
20 years
20. god.
S6-36 1,37 (0.18)
1
c
2
1,96 (0.48) c 4,83 (0.61) bc 6,46 (0.53) c 5,93 (0.56) c 7,11 (0.18) c 7,64 (0.16) d
NS1.3 1,16 (0.27) c 1,87 (0.41) c 5,99 (1.74) ab 10,51 (1.76) a 14,87 (1.59) a 17,9 (0.96) a 22,22 (0.57) a
NS11-8 1,94 (0.08) b 3,89 (0.22) ab 7,38 (0.83) a 8,94 (0.88) b 11,88 (1.99) b 12,13 (2.97) b 13,27 (3.46) c
Pannonia 2,76 (0.24) a 4,42 (0.29) a 7,3 (0.59) a 6,33 (0.14) c 6,49 (1.36) c 11,43 (1.57) b 14,36 (1.96) c
PE19/66 1,46 (0.24) c 3,25 (0.57) b 3,59 (0.58) c 5,99 (1.11) c 6,81 (1.47) c 16,76 (2.09) a 22,04 (2.59) a
S6-7 2,5 (0.32) a 3,99 (0.59) a 3,42 (0.82) c 8,18 (0.61) b 11,41 (1.11) b 16,94 (1.04) a 18,34 (1.21) b
F 30.65*** 23.56*** 13.51*** 13.31*** 26.83*** 23.97*** 31.58***


1 Values in parentheses represent the standard deviation – Vrijednosti u zagradi predstavljaju standardnu devijaciju.


2 The same letters indicate that there is no statistically significant differences between the clones tested by least significant
differences at the risk level of 0.05 – Ista slova znače da ne postoje statistički značajne razlike između klonova po testu
najmanje značajne razlike na razini rizika od 0,05.


nsnon significant –nije signifikantno; * significant at the risk level of 0,05 –signifikantno na razini rizika od 0,05;
** significant at the risk level of 0,01 –signifikantno na razini rizika od 0,01; *** significant at the risk level of 0,001


–signfikantno na razini rizika od 0,001.


Table 8
Mean values of shape parameter (c) of the Weibull diameter structure model and the results of the analysis of
variance test and LSD test at the risk level of 0.05 of study clones per years of measurement.


Tablica8.Srednje vrijednosti parametra oblika (c) modelaWeibull-ove distribucije prsnih promjera i rezultati testa analize
varijance i testa NZR na razini rizika od 0,05 istraživanih klonova po godinama izmjere.


Clone
Klon
Plantation age after planting –Starost nasada nakon sadnje
1 year
1. god.
2 years
2. god.
5 years
5. god.
8 years
8. god.
12 years
12.god.
17 years
17. god.
20 years
20. god.
S6-36 2,235(0.20)1 b2 2,044 (0.27) b3,714(0.59)abc 4,373 (1.37) a 2,971 (0.45) c 3,618 (0.86) b 3,519 (0.81) b
NS1.3 2,084 (0.57) b 2,513 (0.68) b 4,146(1.39) ab 4,963 (1.06) a 5,398 (1.09) a 5,131 (1.57) b 5,123(0.99) ab
NS11-8 3,099 (0.95) b 4,451 (1.03) a 4,831(1.16) ab 4,74 (1.56) a 4,415(1.18) ab 3,556 (0.80) b 3,493 (0.64) b
Pannonia 5,378 (1.90) a 5,133 (0.93) a 5,04 (0.98) a 4,077 (0.46) a 3,425(0.70) bc 5,539(1.44) ab 5,712(1.42) ab
PE19/66 2,95 (0.40) b 4,961 (0.58) a 3,53 (0.59) bc 3,372 (0.19) a 3,228(0.30) bc 5,618(0.70) ab 6,563 (1.07) a
S6-7 4,677 (1.00) a 4,175 (0.89) a 2,378 (0.79) c 4,949 (1.30) a 5,69 (1.46) a 7,986 (3.86) a 7,287 (3.09) a
F 7.12*** 11.23*** 4.08* 1.23ns 5.88** 2.98* 3.96*


1 Values in parentheses represent the standard deviation – Vrijednosti u zagradi predstavljaju standardnu devijaciju.


2 The same letters indicate that there is no statistically significant differences between the clones tested by least significant
differences at the risk level of 0.05 – Ista slova znače da ne postoje statistički značajne razlike između klonova po testu
najmanje značajne razlike na razini rizika od 0,05.


nsnon significant –nije signifikantno; * significant at the risk level of 0,05 –signifikantno na razini rizika od 0,05;
**significant at the risk level of 0,01 –signifikantno na razini rizika od 0,01; *** significant at the risk level of 0,001 –
signifikantno na razini rizika od 0,001.


The differences between the parameters of the Wei-gnificant difference test at the 5%risk level grouped
bull diameter structure model of poplar clones per the study clones in several groups.
years of measurement were significant, and the least si




ŠUMARSKI LIST 11-12/2009 str. 35     <-- 35 -->        PDF

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


Plantation age affects the trend of parameters of the
diameter structure model. In the initial period of plantation
development (five years) there was a sharp increase
in the location parameter (a), after which its
trend changed depending on the clone. In the initial period,
the increase in the scale parameter (b) of theWei-
bull diameter structure model was lower and also the
differences between the clones were lower. The increase
in parameter “b” in the following period was
considerably greater in the majority of the clones, but
the clone S showed the trend of a very slow increase.


6-36


The differences between the study clones were the lowest
in the shape parameter (c).


Diagram 3 Changes in the mean values of the parameters of the Weibull diameter structure model depending on plantation age.


Grafikon 3. Promjena srednjih veličina parametara modela debljinske strukture po Weibull-u u zavisnosti od starosti nasada


4.2.3. Construction of the Weibull diameter structure model


Konstrukcija modela debljinske strukture po Weibull-ovoj distribuciji


Diagram 4 presents the models of diameter structu-Aiming at the objective assessment of similarities
res of the study clones per years.The Diagram shows and differences between the Weibull diameter structure
the changes in the shape of diameter structure with models of poplar clones, the non-parametric Kolmogoadvancing
age, as well as the differences between the rov-Smirnov test was applied at the plantation age of
clones. It can be concluded that some clones had simi-17 and 20 years (Table 9). The test results showed silar
models of diameter structure, especially in the pe-gnificant differences only between diameter structure
riod after the age of 12, which indicates the possibility models of the clone PE 19/66 and the clones S , NS ,


6-7 1-3
th


of their grouping in the definition of management pro-
S in the 17 year, and between the clone PE 19/66


6-36
th
6-7 1-3 6-36


cedures.
and clones S , NS , S and Pannonia in the 20
year.


Table 9 Value of |D| statistics by Kolmogorov-Smirnov test and the comparison of differences in the Weibull diameter


th th


structure models in the 17 and 20 year of poplar plantation age.


Tablica 9.Vrijednosti |D| statistike po testu Kolmogorov-Smirnova i usporedba razlika modela debljinskih struktura po
modelu Weibull-a u 17. i 20. godini starosti nasada istraživanih klonova topola.


Clone –Klon 20 years –20. godina
S
6-36
NS
1-3
NS
11-8
Pannonia PE 19/66 S
6-7
17 years
S
6-36
-0.2039
ns
0.28
ns
0.1454
ns
0.6623
**
0.18
ns
NS
1-3
0.1993
ns
-0.1925
ns
0.195
ns
0.5266
**
0.1941
ns
NS
11-8
0.2626
ns
0.1963
ns
-0.1718
ns
0.3923
ns
0.35
ns
17. godina Pannonia 0.1771
ns
0.24
ns
0.1529
ns
-0.5641
**
0.2641
ns
* * ns ns ***
PE 19/66 0.5758 0.4972 0.3131 0.4661 -0.7144
S
6-7
0.2525
ns
0.1988
ns
0.3939
ns
0.3875
ns
0.6869
***
-


nsnon significant– nije signifikantno;
*significant at the risk level of 0,05– signifikantno na razini rizika od 0,05;
**significant at the risk level of 0,01– signifikantno na razini rizika od 0,01;
***significant at the risk level of 0,001


–signifikantno na razini rizika od 0,001.




ŠUMARSKI LIST 11-12/2009 str. 36     <-- 36 -->        PDF

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


Diagram 4 Models of diameter structure of the study clones depending on plantation age.


Grafikon 4. Modeli debljinske strukture istraživanih klonova u zavisnosti od starosti nasada.


Table 10 Mean values of the percentage of the number of trees with diameters at breast height above 40 cm (N [%])


d>40cm


and the results of the analysis of variance test and LSD test at the 5%risk level.


Tablica 10.Srednje vrijednosti učešća broja stabala prsnih promjera većih od 40 cm (Nd>40cm[%]) i rezultati testa analize
varijance i testa NZR na razini rizika od 5%.


Age
Clone –Klon
S
6-36
NS
1-3
NS
11-8
Pannonia PE 19/66 S
6-7
F
1
Starost Nd>40cm [%]
17 years 0.0 c
2
1.4 bc 4.4 ab 0.4 bc 13.3 a 0.0 c 5.31
**
20 years 1.0 d 11.3 bc 19.1 b 4.5 cd 50.0 a 0.8 d 12.1
***


1


The comparison was preceded by the transformation arcsin aiming at the homogenisation of the variances


(%Nd>40cm)1


–Usporedba je izvršena uz prethodnu transformaciju arcsin (%Nd>40cm)1u cilju homogenizacije varijanci.


2


The same letters indicate that there is no statistically significant differences between the clones tested by least significant


differences at the risk level of 0.05 – Ista slova znače da ne postoje statistički značajne razlike između klonova po testu naj


manje značajne razlike na razini rizika od 0,05.


ns* **


non significant –nije signifikantno; significant at the risk level of 0,05 –signifikantno na razini rizika od 0,05; signifi


***


cant at the risk level of 0,01 –signifikantno na razini rizika od 0,01; significant at the risk level of 0,001 –signifikantno
na razini rizika od 0,001.




ŠUMARSKI LIST 11-12/2009 str. 37     <-- 37 -->        PDF

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


The practical significance of the obtained models is
best seen by the example of estimating the percentage
of the number of trees with diameters at breast height
above 40 cm.The diameter at breast height above 40 cm
makes it possible to produce the best quality veneer
logs (Pudar, 1986; Krznar,1987), so the share of
such trees indicates indirectly the plantation value.
Table 10 presents the mean relative percentage of the
number of trees with diameters at breast height above


40 cm from the diameter structure model and the results


of the analysis of variance test and LSD test at the 0.05
risk level.The analysis of variance points out the signi


th


ficant differences between the clones, both in the 17


th


and in the 20 year. Significantly the highest percentage
of the number of trees with diameters at breast height
above 40 cm was attained by the clone PE 19/66, and
the lowest percentage by the clones S and S .


6-7 6-36


5. DISCUSSION – Rasprava


The researched plantation of six newly selected clones
showed different values of stand quadratic mean
diameter (dg) and total basal area per hectare (G) at the
end of the study period (20 years). Based on the results
of LSD test at the 5%risk level, the clones were grouped
in several production groups.Also, the periodical
measurements of diameter at breast height showed different
diameter growth of the clones depending on the
age. In the initial period, the clonesPopulus deltoides
Bartr. ex Marsh. (S , NS , PE 19/66, NS , S ) developed
more intensively than the clone Pannonia (Populus
× euramericana (Dode) Guinier). In the later
period, Pannonia developed more intensively and the
clones ofP.deltoidesBartr. ex Marsh. showed the differentiation.
The different growth characteristics of the
clonesP.deltoidesBartr. ex Marsh. and P.×euramericana(
Dode) Guinier, prevent the reliable productivity
differentiation of the clones before the ages of 16–18
years at plantation density of 400 trees per hectare (Andrašev,
2008). Taking into account the above facts, the
differentiation of study poplar clones by total basal
area per hectare (G) and stand quadratic mean diameter
(dg) at the ages of 17 and 20 years showed (LSD test)
that clone PE 19/66 can be classified in one group, and
the other clones in the other group. The constructed
Weibull models of diameter structure, and the derived
percentage of the number trees with diameters at breast
height above 40 cm (Nd>40cm [%]) confirmed thegrouping
of the clones in two groups, as well as based on
growth elements (G,dg), which point out their implementation
in productive differentiation.


The study results refer to six black poplar clones,
SectionAigeiros(Duby), four of which were registered
as cultivars in Serbia, and the other two are still undergoing
the selection procedure.The registered poplar clones
(S , NS , NS and Pannonia) attained similar


6-7 1-3 11-8 6-36


6-36 1-3 11-8


plantation growth elements (dg ,G), and diameter structure,
which was confirmed by statistical tests.The other
two clones which were in selection procedure (S and


6-7


PE 19/66) attained significant differences in growth elements
and plantation structure.Clone S did not have


6-7


significantly lower values of growth elements and
structure compared to registered clones. However,
clone PE 19/66 attained a significant advantage in the
elements of growth and structure compared to the registered
clones at the plantation age of 20 years, which,
according toMarkovićet al.(1997) andAndrašev
(2008), can be taken as the rotation period for the density
of 400 trees per hectare, so this clone is a reliable
candidate for a soon registration and putting in mass
production.


The study results indicate that theWeibull diameter
structure model can be successfully applied in the estimation
of diameter structure of the newly selected poplar
clones at different plantation ages.


The application of the model of theWeibull three-
parameter distribution, especially the calculation of the
location parameter (a), was made difficult and it was
evaluated in different ways: using different mathematical
expressions (Zarnoch andDell,1985), fixed values
of location parameter 0 or d , the percentiles of


min


minimal diameter, with frequent value 50%·d (Bai


min


ley and Dell, 1973, Knoebel, et al, 1986, Lei,
2008). Our research indicates that the choice of location
parameter (a) inWeibull distribution should not be
uniformly defined, and that further research is necessary
aiming at reliable methods of diameter structure
modelling in poplar plantations.


Taking into account the so-called “biological”interpretation
of the calculated parameters, it can be concluded
that location parameter (a) is the minimal diameter
in the plantation (but not also in the sample based on
which it is predicted). In most clones, location parameters
depending on the age show an increasing trend; in
the initial period the trend is sharp (till the age of 5),
and later on it is slower or more intensive, depending
on the clone.The above can be related to the so-called
“solitary growth”in the initial period before crown closure
(from fifth to eighth year) and the so-called “stand
growth”with the competitive impact of trees.The observed
significant drop of location parameter (a) in the
clone PE 19/66 is the consequence of the applied method
and it shows that in periodical measurements the
so-called “biological component”should be “incorporated”
in its definition.


The scale parameter (b) is in high correlation
(R=0.884) with the variability of diameters at breast




ŠUMARSKI LIST 11-12/2009 str. 38     <-- 38 -->        PDF

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


height, represented by standard deviation (sd). Its
change depending on plantation age, with a high value
of the correlation coefficient (R=0.74), points to the
differentiation process of the trees in the plantation.


The parameters of the Weibull diameter structure
model (a,b,c) of the study poplar clones, according to
the analysis of variance F-ratio, decrease till the age of
8, and then they increase.Asimilar F-ratio trend was
shown by the stand quadratic mean diameter (dg) and
total basal area per hectare (G), with the minimal F-
ratio at the age of 12.As the parameters describe the internal
plantation structure, the earlier change in
parameters compared to growth elements indicates the
growth changes in the trees of different categories (diameters),
which enables the adequate evaluation of the
plantation state and future development.


The quantification of similatities and differences in
the location, scale and shape of diameter structure during
the development of the plantations of different clones
contributes to the advanced study of plantation
development and can be applied in the construction of
the growth model of selected clones, and their production
differentiation.


The assessment of the effects of clone and plantation
age on diameter structure enables a more reliable
construction and estimation of the future structure, and
also its incorporation in growth models.


6. CONCLUSIONS – Zaključci


Based on the research of 20-year old test plantation
consisting of several newly selected clones of black poplar,
Section Aigeiros(Duby), which are either registered
or are in the selection procedure, we can conclude
as follows:


– there are significant differences in growth elements
(dg, G) between individual clones in the test plantation
at the end of the study period of 20 years, which
provides the basis for their production differentiation;
– periodical measurements of diameters at breast
height showed different growth of the clones depending
on the age: in the initial period of development,
the clonesPopulus deltoidesBartr. ex Marsh. (S ,


6-7


NS , PE 19/66, NS , S ) developed more inten


1-3 11-8 6-36


sively than the clone Pannonia (Populus×euramericana(
Dode) Guinier), and later on Pannonia had a
more intensive growth, while the clones ofP.deltoidesBartr.
ex Marsh. showed the differentiation;


– theWeibull distribution model is suitable for diameter
structure modelling of the study poplar clones at
different plantation ages, and the applied method of
predicting the location parameter (a) of theWeibull
distribution model showed that in 90.6 % of the
study sample, the parameter “a”ranged between 50
and 90%, and in 52.4% “a”ranged from 80 to 90%
of the minimal diameter;


– with higher plantation ages, location parameter (a)
and scale parameter (b) also increased with small
oscillations, which was confirmed by the significance
of the correlation coefficient of 0.71 and 0.73
respectively. This was shown by the shift of the
curve of diameter structure model to the right, towards
larger diameters, and in a wider range of diameters
at breast height with a lower relative
frequency of the modal degree;

in the initial period, F-ratio of all three parameters of
theWeibull diameter structure model decreased and
reached the minimal value at the age of eight, and the
predominantly increasing trend started at the age of
twelve, which points to the changes in diameter
structure of the clones depending on the age;

the constructed models of poplar clone diameter
structure show the clone grouping in two groups,
which was confirmed by the non-parametric Kolmogorov-
Smirnov test, the analysis of variance test, and
LSD test for the percentage of the number of trees
with diameters at breast height above 40 cm. This
emphasises the possibility and the need of their grouping
in the definition of management procedures.


7. REFERENCES – Literatura


Anderson,T.W.,D.A.,Darling,1954: ATest of
goodness-of-fit. Journal of theAmerican StatisticalAssociation,
49: 765–769,Alexandria, USA.


Andrašev,S., S.Rončević, M.Bobinac,2003:
Uticaj gustine sadnje na debljinsku strukturu
klonova crnih topola S 6-7 i M-1 (SekcijaAigeiros
(Duby)). Glasnik Šumarskog fakulteta, 88:
7–16, Beograd.


Andrašev,S., M. Vučković, S.Rončević, M.
Bobinac,2004: Mogućnost modelovanja debljinske
strukture i izračunavanje zapremine za


sada klonova crnih topola. Glasnik Šumarskog


fakulteta 90: 37–51, Beograd.


Andrašev,S., M. Vučković, S.Rončević, M.
Bobinac,2006: Modeli rasta stabala crnih topola
sekcije Aigeiros (Duby). Glasnik Šumarskog
fakulteta 94: 81–90, Beograd.


Andrašev,S., 2008: Razvojno proizvodne karakteristike
selekcionisanih klonova crnih topola (sekcija
Aigeiros Duby) u gornjem i srednjem




ŠUMARSKI LIST 11-12/2009 str. 39     <-- 39 -->        PDF

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


Podunavlju. Disertacija, Šumarski fakultet Uni-Forest Science, 54 (12): 566–571,Prague, Czech
verziteta u Beogradu, p. 427. Republic.


Bailey, R.L., T.R.,Dell,1973: Quantifying diame-Marković,J., Z. Pudar, S. Rončević,1995:
ter distributions with the Weibull function. Fo-Zna čaj proizvodnje drveta topola i vrba kao sirorest
Scienece 19 (2): 97–104, Bethesda, USA. vinske osnove za industriju celuloze i papira u


Jugoslaviji. Radovi Instituta za topolarstvo, br.


Conte, S. D, C.deBoor,1980: Elementary numeri26:
5–19, Novi Sad.


cal analysis, algorithmic approach. Third edition.
McGraw-Hill Book Company, p 432, New
Marković,J., S. Rončević, Z. Pudar,1997:
York.
Izbor razmaka sadnje pri osnivnju zasada topola.


Topola, 159/160: 7–26, Novi Sad.


......, .. .., 1984: ............. ........ ..
........ . ....... ....... .........


Pudar,Z., 1986: Ekonomski aspekti proizvodnje dr...........,
2: 65–70, ......, .....
veta topole, Populus ×
euramericana (Dode)
Kotar, M., 2005: Zgradba, rast in donos gozda na


Guinier, cl. I-214 u zasadima različite gustine.
ekoloških in fizioloških osnovah. Zveza gozdar-


Radovi Instituta za topolarstvo, 17: 1–121, Novi
skih društev Slovenije in Zavod za gozdove Slo-


Sad.
venije, p. 500. Ljubljana.


StatSoft Inc., 2006: STATISTICA(data analysis soft-
Knoebel, B. R., H. E. Burkhart, D. E. Beck,


ware system).Version 7.1.
1986: A growth and yield model for thinned


Škorić,A., G.Filipovski, M.Ćirić,1985: Klastands
of yellow-poplar. Forest-Science-Mo no


si fikacija zemljišta Jugoslavije. Akademija nau graph
27. p. 62, Bethesda, USA.


ka i umjetnosti Bosne i Hercegovine; Posebna
Knowe, S.A., G. S.Foster, R. J.Rousseau,W.


izdanja, knjiga LXXVIII; Odeljenje prirodnih i


matematičkih nauka, knjiga 13: 1–72, Sarajevo.
xing study: predicted diameter distributions.


L.Nance,1994: Easter cottonwood clonal mi-


Weibull,W.,1951:Astatistical distribution function
Can. J. For. Res, 24: 405–414, Ottawa, Canada.


of wide applicability. Journal of applied mechaKrznar,
A., 1987: Utjecaj debljinske strukture na vri


nics, 18: 293–297, NewYork, USA.
jednost sastojine. Šumarski list, 10–12: 631–344,


Zarnoch, S.J., T.R.Dell,1985:An evaluation of
Zagreb.


skewness and maximum likelihood estimators of
Lei,Y.,2008: Evaluation of three methods for estima-


Weibull parametrs. Forest Science, 31: 260–268,
ting theWeibull distribution parameters of Chi-


Bethesda, USA.
nese pine (Pinus tabulaeformis). Journal of


SAŽETAK: Cilj rada je utvrditi pogodnost Weibull-ove distribucije za konstrukciju
modela debljinske strukture više klonova topola sekcije Aigeiros
(Duby), uz primjenu tzv. “hibridnog sustava” nalaženja parametara modela
iz uzorka, pri čemu se za nalaženje nepoznatog parametra lokacije (a) modela
koristi više percentila minimalnog promjera u rasponu od 0÷dmin. Također, cilj
rada je ispitati promjenu parametara modela u zavisnosti od starosti nasada i
klona topole, kao i njihov odnos s elementima rasta nasada (dg, G).


Istraživanja su obavljena u pokusnom nasadu starom 20 godina, koji se sas toji
od više klonova (sorti) crnih topola sekcije Aigeiros (Duby): S6-36, NS1-3,
NS11-8, Pannonia, PE 19/66 i S6-7. Nasad je osnovan na zemljištu tipa fluvisol,
pjeskovito-ilovaste forme, pri razmaku sadnje od 5 × 5 m (400 stabala po hektaru),
sa sadnicama tipa 2+0. U pokusnom nasadu svaki klon ima četiri reda
(ponavljanja) i po 20–25 biljaka u svakom redu. U pokusnom nasadu su periodično
mejreni prsni promjeri svih stabala (s točnošću od 1 mm), nakon prve,
druge, pete, osme, dvanaeste, sedamnaeste i dvadesete godine od osnivanja.
Mjereni prsni promjeri stabala u svakom redu, za svaki istraživani klon i godinu
izmjere, predstavljali su uzorak stabala za konstrukciju modela debljinske struk ture
(ukupno 168 uzoraka). Za svaki uzorak stabala izračunata je temeljni ca,
kao zbroj temeljnica svih stabala, te izračunat srednji promjer po temeljnici.


Kao model izabrana je Weibull-ova distribucija s tri parametra, čija je
funkcija gustoće definirana izrazom (1), a kumulativne distribucije izrazom




ŠUMARSKI LIST 11-12/2009 str. 40     <-- 40 -->        PDF

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


(2). Za nalaženje nepoznatih parametara Weibull-ove distrbucije korišten je
tzv. “hibridni sustav”, odnosno metoda momenata u kombinaciji s metodom
percentila (Knoebel, et al, 1986). Parametar položaja (a) dobiven je po metodi
percentila, pri čemu su korišteni sljedeći percentili minimalnog promjera:
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. Parametri skaliranja


(b)i oblika (c) dobiveni su po metodi momenata, pri čemu je za njihovo nalaženje
izvršeno oduzimanje mjerenih veličina (prsnih promjera) od prethodno definiranog
parametra “a”. Ova metoda zasniva se na jednadžbama prvog (x-) i
drugog (x–2) običnog momenta Weibull-ove dvoparametarske distribucije (5,
6). Procijenjena varijanca (s2) Weibull-ove distribucije definirana je izrazom
(7), a koeficijent varijacije (cv) izrazom (8). Nalaženjem prvog i drugog običnog
momenta, kao i koeficijenta varijacije iz uzorka i stavljanjem u formulu 8,
koeficijent varijacije je funkcija samo jednog parametra (c), te se može dobiti
postupkom iteracije. U radu je korištena kombinacija metode sekante i metode
polovljenja intervala uz unaprijed zadanu točnost od 10-6(Conte i de Boor,
1973). Dobivena veličina parametra „c“ poslužila je da se dobije veličina parametra
“b” iz relacije (9). Za svaki od navedenih percentila u svakom ponavljanju
dobiven je parametar „a“ modela, a metodom momenata parametri
“b” i “c”. Zatim je izvršena usporedba empirijske debljinske strukture i modela
primjenom Anderson-Darling statistike (A2) (Anderson i Darling, 1954)
po formuli (10). Izbor percentila minimalnog promjera izvršen je na temelju
minimalne veličine A2statistike za sva 4 ponavljanja u okviru istog klona i starosti
nasada. Za svaki od izabranih percentila minimalnog promjera u okviru
svakog klona i starosti nasada izvršeno je ponovno nalaženje sva tri parametra
Weibull-ove distribucije za svako ponavljanje. Stupanj slaganja modela debljinske
strukture i empirijske distribucije izvršen je neparametarskim testom
Kolmogorov-Smirnova, nalaženjem |D| statistike (11). Dobiveni parametri
Weibull-ove distribucije po pojedinim ponavljanjima korišteni su za utvrđivanje
razlika između istraživanih klonova u pojedinim starostima (godina nakon
sadnje), pri čemu je korišten statistički test analize varijance i test najmanje
značajne razlike (NZR), na razini rizika od 5 %.


Istraživani klonovi ostvarili su značajne razlike u elementima rasta (dg, G)
na kraju istraživanog razdoblja od 20 godina, što pruža osnovu za njihovo
proizvodno diferenciranje. Na osnovi periodičnih izmjera prsnih promjera
utvrđen je različit rast istraživanih klonova u debljinu, u zavisnosti od starosti:
u početnom razdoblju razvoja klonovi Populus deltoides Bartr. ex Marsh.
(S6-7, NS1-3, PE 19/66, NS11-8, S6-36) rastu intenzivnije od klona Pannonia (Populus
× euramericana (Dode) Guinier), a kasnije klon Pannonia ima intenzivniji
rast, dok između klonova P. deltoides Bartr. ex Marsh. dolazi do
međusobnog diferenciranja. Tijekom cijelog razdoblja od 20 godina klon PE
19/66 ostvario je najveće veličine, kako srednjeg promjera po temeljnici (dg),
tako i ukupne temeljnice po hektaru (G) i izdvaja se od ostalih klonova po
testu NZR na razini rizika od 5 % (tablice 1, 2, grafikon 1).


Model Weibull-ove distribucije pokazao se pogodnim za modeliranje debljinske
strukture istraživanih klonova topola u različitim starostima nasada, a
primijenjena metoda nalaženja parametra položaja (a) modela Weibull-ove
distribucije pokazala je da se u 93,1 % istaživanog uzorka parametar „a“ nalazi
o rasponu od 50–90 %, a u 54,2 % u rasponu od 80–90 % minimalnog
promjera uzorka (grafikon 2). Usporedbom modela kumulativne distribucije i
empirijske kumulativne distribucije neparametarskim testom Kolmogorov-
Smirnova, povrđena je sličnost kod svih 168 uzoraka.


Uz male oscilacije s povećanjem starosti nasada povećavaju se parametri
položaja (a) i skaliranja (b), što je potvrđeno koeficijentom korelacije od 0,71
i 0,73 (grafikon 3). Tose manifestira u pomicanju krivulje modela debljinske




ŠUMARSKI LIST 11-12/2009 str. 41     <-- 41 -->        PDF

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


strukture udesno k većim promjerima i u širem rasponu prsnih promjera s manjom
relativnom frekvencijom modalnog stupnja (grafikon 4). Promjene para-
metra oblika raspodjele (c) manje su izražene, što potvrđuje iznos koeficijenta
korelacije od 0,36 (grafikon 3). Međutim, njegovo značenje na razini rizika od


0.001 ukazuje na trend povećanja sa starošću nasada, odnosno na promjenu
oblika debljinske strukture.


U početnom razdoblju koda sva tri parametra modela debljinske strukture
po Weibull-u F-količnik ima trend opadanja i dostizanja minimalne vrijednosti
u osmoj godini, a u dvanaestoj godini pokazuje porast, odnosno pretežito
trend povećanja, što ukazuje na promjene u debljinskoj strukturi istraživanih
klonova u zavisnosti od starosti.


Konstruirani modeli debljinske strukture istraživanih klonova topola pokazuju
grupiranje klonova u dvije grupe, što je potvrđeno neparametarskim
testom Kolmogorov-Smirnova, kao i testom analize varijance i testom NZR za
učešće broja stabala prsnih promjera debljih od 40 cm, te ukazuje na mogućnost
i potrebu njihovog grupiranja pri definiranju odgovarajućih gospodarskih
postupaka


Ključne riječi:crna topola, klonovi, debljinska struktura, Weibulova
funkcija gustoće