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The study on the structure of Norway spruce old−growth stands of original provenience in upper mountain zone was carried out in a nature reserve ‘Śnieżnik Kłodzki', located in the Lądek Zdrój Forest District (SW Poland). The fieldwork was conducted in three spruce stands located at the altitude of 1,215 and 1,235 m a.s.l. The breast height diameters (dbh) of living trees were characterized. The aim of this study is to characterize the structure of dbh of trees in spruce stands of subalpine forest. Implementation of the objective will be associated with the characteristics of empirical distributions dbh and an indication of theoretical distributions of continuous random variable best approximating the structure of dbh. The comparison of empirical dbh distributions with 36 theoretical distributions were carried out, but goodness−of−fit tests proven statistically significant compatibility with seventeen of them: beta (4−parameter), Cauchy, exponential power, folded normal, gamma (3−parameter), generalized logistic, Laplace, logistic, loglogistic, loglogistic (3−parameter), lognormal (3−parameter), noncentral chi−square, normal, smallest extreme value, triangular, Weibull and Weibull (3−parameter). The dbh distributions in upper mountain spruce stands are the best fitted with logistic distribution, and subsequently loglogistic (3−parameter), generalized logistic and Weibull (3−parameter) distributions. That four distributions can be used in individual tree growth models while generating structure dbh of trees in the stand. The knowledge of dbh structure in protected stands could be helpful in shaping spruce stand structure in planned silvicultural treatments, and shows the need for silvicultural treatments in upper mountain spruce stands.
Modelling is one of the basic activities in the field of biological sciences. The main purpose of modelling is to illustrate relationships and phenomena of biological objects (and traits describing them) and present reality as close as it is possible with expected range of error. In the paper we presented results of approximation of the selected theoretical distributions, i.e. singular (normal, log−normal, gamma, two− and three−parameter Weibull, Johnsons SB) and finite mixture (normal, log−normal, gamma and Weibull) to tree diameter distributions in beech stands (Fagus sylvatica L.). The main aim of the study was to evaluate goodness of fit of selected distributions, prove legitimacy of using mixture distributions in even−aged, single−species beech stands and find the best distribution of model diameter distribution in a whole production cycle of this kind of stands. The study was conducted in 32 sample plots located in north−western Poland in stands at the age between 9 and 140. For the evaluation χ² test, a modified error index (e) and a root mean square error (RMSE) were used. We found that the RMSE and e are related in most cases with stand density. Mean values of used indicators show that finite mixture distributions are better in comparison to singular distributions. The best fitted distributions were Johnsons SB and mixture of gamma ones. Range of error index expressed with number of trees per hectare ranges 16.5−32.8% and 7.7−27.0% for this distributions, respectively. Conversely the worst results were obtained by fitting two−parameter Weibull distribution both in singular and mixture. Future diameter distribution model based on the probability distribution should be based on Johnson SB or 3−parameter Weibull distribution from singular ones or gamma from finite mixture. In case of age class apportionment unequivocal the best result were obtained with Johnson SB and partially with mixture gamma distributions.
We evaluated the usefulness of ten theoretical probability density functions for the approximation of the breast height diameter distributions in managed alder stands growing in the western part of the Sandomierz Basin (southern Poland). The study material consisted of measurement results for 22,530 alders obtained for 844 circular sample plots, established in 163 stands aged 6−89. In the area of particular stands, from 2 to 10 age−dependent circular sample plots sized 0.01−0.10 ha were systematically arranged. We evaluated the following theoretical probability distributions: normal (N), double normal (P−N), Johnson SB (J), Weibull (W), beta (B), Burr (Bu), gamma (G), log−normal (L−N), log−logistic (L−L) and Birnbaum−Saunders (B−S). Additionally the usefulness of the W, B, Bu, G, L−N, L−L and B−S distributions with omitted location parameter was tested. The goodness−of−fit of each probability density function was assessed based on the Kolmogorov−Smirnov statistic, the root mean square error and ranks established on their basis. It was found that DBH distribution of majority stands is characterized by right−sided asymmetry and negative kurtosis. It was showed that the best distribution for the breast height diameter structure approximation of alder stands is the Johnson SB (J) distribution. To describe the frequency of thickness, normal (N) and log−logistic (L−L) distribution should not be used. Elimination of location parameter significantly affects usefulness of probability density functions to modeling empirical distributions of breast height diameter. Gamma and Burr distributions with omitted location parameter gave the best results in description the frequency of breast height diameter. In certain situations, these distributions can also be used to describe the structure of breast height diameter of alder stands.
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