Multivariate beech stands structure in the north-east mega slope of the Ukrainian Carpathian: theoretical aspects and practical significance

High productivity even-aged beech stands of the north-east mega slope of the Ukrainian Carpathian in different age are the study object in this scientific work. The modelling of diameters and heights frequencies distribution performed taking into consideration their relations in dynamic for the detailing of infor-mation about the beech forest structure in the study region. The initial data were received by the continuous inventory method. This data includes measure of di-ameters and heights of beech trees. The age range of investigated stands was from 10 to 120 years. The main parameters of one-dimensional distribution model are the mathematical expectation, standard deviation, skewness and kurtosis. When more complicate system is studied he model getting more complicate and the number of parameters, that describe it, is growing. Therefore, for modelling of diameter and height distribution, taking into account the relations between them, it is necessary to employ from 5 to 14 parameters. The equation of fitted surface type A with 14 parameters was chosen as the model of diameters and heights distribution. The algorithm of modelling theoreti-cal surface has the evaluation of such parameters of diameter and height distribu-tion: position (mean and mean-square parameters), scale (standard deviations), form (skewness and kurtosis), relation (main mixed moments of 1-3 degree). It is determined, that the standard deviation of tree’s count deviation by height are increasing from 1,1…0,9 m in 10-year-old stands to 5,7…4,4 m at 120-year-old stands. The standard deviation of tree’s count deviation by diame-ters are increasing from 1,6…1,2 cm in 10-year-old stands to 15,9…12,4 m at 120-year-old stands. The skewness of tree’s count deviation by height are in-creasing from -0,46…-0,36 in 10-year-old stands to -0,78 at 120-year-old stands. The skewness of tree’s count deviation by diameter are decreasing from 0,86…1,02 in 10-year-old stands to 0,33…0,32 at 120-year-old stands. The kur-tosis of tree’s count deviation by height are decreasing from -0,53…-0,69 in 10-year-old stands to 0,12…0,13 at 120-year-old stands. The kurtosis of tree’s count deviation by diameter are decreasing from 0,53…0,88 in 10-year-old stands to -0,43…-0,45 at 120-year-old stands. The coefficient of correlation be-tween the features is in average 0,76. Most veritable values of others main mixed moments are the next: r12=-0,358, r21=-0,109, r13=2,198, r31=2,001, r22=1,905. The height and diameter parameters of beech stands were the model’s in-dependent variables of dynamic of state parameters. The average height of stand is a function of top height. The average age and productivity class of stands are the model’s independent variables of dynamic of form. Complex mixed moments were evaluated through the variables of more simple parameters. The result of these modelling are the dynamic standards of two-dimensional distribution of quantity of trees by height and diameter for the beech stands of Ic…I classes of productivity. The modelling of multi-dimensional relations allows to solve the practical tasks. The stands volume calculation was performed using a multivariate distri-bution model, which allows expanding the information base of the assortment and commodity structures. Application of the method of multivariate estimation of diameters and heights distribution in different ages, forest types and different productivity classes give an opportunity to formalize projecting of thinning. The study for estimation of multivariate random models gives an oppor-tunity to link the stem, underground and above-ground parts of the stands. This study will assist in developing a complete conception of the forest with a multi-plicity of dimensions and relations