Distribution of Tree Diameters in Mixed-Age Stands of Small-Leaved Linden
DOI:
https://doi.org/10.37482/0536-1036-2026-4-20-34Keywords:
diameter at a height of 1.3 m, diameter distribution, diameter modeling, finite mixture model, skew normal distribution, small-leaved linden, Tilia cordata Mill., Republic of BashkortostanAbstract
Identification and assessment of quantitative parameters of forest stands are initial requirements for planning strategies and options for forestry measures. These parameters include structural diversity, which can be determined by modeling the density of distribution probability of tree diameters at breast height of 1.3 m (dbh). The study aimed at describing the tree stand structure and selecting the most appropriate finite mixture model (FMM) of the empirical diameter distribution of natural mixed-age small-leaved linden (Tilia cordata Mill.) stands in the central-eastern part of the Republic of Bashkortostan. We carried out a hierarchical cluster analysis using Ward’s method on tree parameters obtained from 19 sample plots, and determined the suitability of two-component finite mixture methods based on the gamma distribution, Weibull distribution, log-normal distribution, and skew-normal distribution. In order to fit the FMM to the diameter data, we used an expectation-maximization algorithm implemented in the ForestFit package, which was developed for the R environment. All the stands studied were classified as unimodal; the diameter distribution curves were described as skewed, complex, and irregular; no bimodal distributions of the M-shaped type with two peaks or downward-sloping inverted J-shaped distributions were observed. It was found that the mixture of the skew-normal model consistently demonstrated the best performance in terms of the Kolmogorov–Smirnov and Anderson–Darling goodness-of-fit statistics, as well as the Akaike and Bayes information criteria, for all sample plots. An assessment of the relative quality ranks of the correspondence allowed the finite mixtures to be ranked as follows: Weibull (3.5 – worst correspondence), log-normal distribution (3.2), gamma (2.9), and skew-normal distribution (1.0). Thus, an FMM based on a skew-normal distribution can reliably approximate the diameter distribution probabilities of small-leaved linden stands of mixed ages; it is highly flexible and offers an effective balance between accuracy and complexity. FMM allow the simultaneous estimation of the proportion and age distribution of trees and can serve as a key tool for converting field measurement data into dynamic models that predict forest development.
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Copyright (c) 2026 Gabdelkhakov A.K., Fazlutdinov I.I., Konovalov V.F., Martynova M.V., Garfutdinova S.I.
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