*2.3. Data Analysis*

Contrary to commonly used frequency distributions based on DBH as a continuous variable [34,35], in this study the focus was on the discrete variable, that is, the tree count data. The frequency distributions of count data are based on the number of sample plots, where each plot contains 0, 1, 2, ... , *n* trees of a particular species, or of all species combined. In this study, the discrete (count) data were divided into understory (≤27.5 cm DBH) and canopy trees (>27.5 cm DBH). The ecological rationale for this division is that live trees with a DBH above 27.5 cm represent "definitive" gap-fillers in Dinaric old-growth forests [36]. Another reason for using 27.5 cm as a dividing value is of a mathematical nature. Namely, DBH categories that are too narrow (e.g., 5 or 10 cm wide) often result in zero values, or such a low number of individuals per category/class that it would prevent the proper application of the chi-square test. It is well known that this test requires at least five individuals per class or category [37]. Thus, the two broader DBH categories were used to "capture" enough trees for robust statistical analysis.

Following this division of trees into the understory and canopy layer, in the next step the dispersion index *Ic* was applied to quadrat (plot) counts per individual species, for conifers combined, and for all trees combined (all species). This index is also called the variance-to-mean ratio as it is based on the relationship of the sample mean to the sample variance [38], and its computation was conducted at the stand level [9]. Theoretically, if index values are equal to 1, then the tree count data are randomly distributed. However, in forest ecosystems it is a rare phenomenon for the mean and variance to be absolutely equal, so small deviations from 1 are still "allowed" for tree count data to be classified as random. Specifically, the *Ic* index is based on the Poisson distribution [11]. Thus, for statistical inferences about significant deviations from 1 (randomness), confidence envelopes were constructed by using a <sup>χ</sup><sup>2</sup> test with *<sup>n</sup>*−1 degrees of freedom, where *<sup>n</sup>* is the number of quadrats (plots). The testing was set at the *p* > 0.05 level. Namely, if the value for χ<sup>2</sup> fell within an envelope between the χ<sup>2</sup> tabular values of 0.975 and 0.025 probability levels, then agreement with a random distribution was reached, indicating that the variance virtually equals the mean. Considering the sample size of 40 plots in each studied stand, the count data in this study were classified as random when their computed *Ic* values fell between 0.68 and 1.42. The computed index values above and below this range denoted a significant deviation from randomness. Specifically, *Ic* values smaller than 0.68 represented an evenly-scattered (regular) distribution of individuals in the population, whereas values above 1.42 indicated a clumped (aggregated) pattern.

With respect to the division of trees into understory and canopy layers, the count data were also modeled per individual species, for conifers combined, and for all trees combined (all species). A Poisson distribution was applied under the assumptions that each sample plot has an equal probability of hosting a tree, the occurrence of a tree in a plot is not influenced by other trees, and the mean number of trees per plot remains constant for all sample plots in a given stand [9].

The Poisson distribution describes the probability *p* of 0, 1, 2, 3, ... , *n* trees occurring in any selected sample plot, while the constant *e* is Euler's number, which equals 2.718282. If the Poisson model was accepted, then a random pattern was confirmed. However, if the Poisson model was rejected, then binomial and negative binomial distributions were employed for regular and clumped patterns, respectively.

$$p\_n = (\lambda^n / n!) \cdot e^{-\lambda} \tag{1}$$

A binomial distribution applies if the probability

$$p(\mathbf{x}) = (N! / (\mathbf{x}! (N - \mathbf{x})!) \cdot p^{\times} \cdot (1 - p)^{N \cdot \mathbf{x}} \tag{2}$$

where *p*(*x*) is the probability of a sample plot containing a specified number of trees *x*, and *N* is the total number of observed trees in sampled plots.

In the negative binomial model, the expected probability of obtaining a given value of a count, *r*, is given by

$$p(r) = \left[ (\Gamma(k+r))/(r!\Gamma(k))\right] \cdot (m/(k+m))^r \cdot (k/(k+m))^k \tag{3}$$

where *p*(*r*) is the probability of getting *r* individuals in the sample plot, *m* is the mean, and *k* is the "shape" parameter. Γ(*k*) is the gamma function of *k*, and it equals Γ(*k*) = [k+1]!

All probabilities were obtained by the recurrence relation [39]. If a negative binomial distribution could not be rejected, then it was concluded that the studied tree species exhibits a clumped pattern. If a binomial distribution could not be rejected, then a regular species distribution pattern was confirmed. The goodness-of-fit of all applied models was tested by applying the χ<sup>2</sup> test, that is, by comparing the observed frequencies with the expected ones, but now with *n*–1–*q* degrees of freedom, where *n* is the number of frequency classes after necessary pooling [37], and *q* is the number of distribution parameters. In each case one degree of freedom was lost due to the overall sum, while additional degrees of freedom were lost depending on the number of distribution parameters.
