**1. Introduction**

Modeling the distribution of tree species in mixed forests has been an important task in forest ecology in the last two decades [1–4]. For this purpose, researchers usually tend to gather both discrete (count) and continuous data on variables of interest in forest ecosystems. However, in many instances, continuous data are limited or not available at all due to financial constraints or because standard inventory procedures encompass only count data. In addition, the focus of the research might solely be species richness for which only count data are necessary.

Count data that originate from different fields of study typically follow a Poisson, negative binomial, or in some cases, binomial distribution [5]. For instance, these distributions have often been used to analyze and model count data in scientific fields such as parasitology [6], veterinary medicine [7], ornithology [8], and estimation of ore reserves [9]. However, their application in the analysis of the distribution of tree species is not very common. Instead, the distribution of tree species in forest research has traditionally been studied by measuring the tree diameter at breast height (DBH), that is, 1.30 m above the ground. Consequently, the conventional approach to analyzing tree species distributions assumes the construction of frequency distributions with DBH classes that usually range from 4(5) to 10 cm. Then, a proper model can be fitted to such grouped data or to the raw data [10]. Such models based on DBH as a continuous variable are still valuable in forest research as they provide

information about species density across a range of DBH classes. However, they do not provide insight into how a species is spatially distributed across an observed area, which is of high importance to ecologists, forest managers, and nature conservationists.

A wide range of data may be used for spatial analysis: the mapped locations of trees in a plane (point pattern process); trees mapped with an associated attribute such as DBH (a marked point process); spatially dispersed sample plots in a systematic or random manner; grids of units, each with quantitative or qualitative characteristics; and so on. Consequently, spatial methods can be classified with regard to the kinds of data to which they are applied. Spatial patterns are commonly divided into random, aggregated (clumped), and regular, whereby they usually correspond to a Poisson, negative binomial, or binomial distribution, respectively [9].

When considering the small-scale spatial level in forest stands, we differentiate between distance-based and angle-based methods and their corresponding indices [11], which are used to quantify and describe spatial patterns of neighboring trees at the "local" level. This small-scale or local level does not usually exceed several hundred square meters and, in practice, often corresponds to the size of small forest inventory plots [12,13]. On the other hand, the quadrat count method may be applied to "global" or "regional" spatial statistics when the goal is to determine the species distribution pattern over the entire study area [9], for instance, at the stand level. It is important to note that the plots used in this method do not have to be square-shaped; they can also be circular or rectangular [14]. The dispersion index frequently "accompanies" quadrat count analysis. The quadrat count method is based on contiguous or scattered quadrats (plots or sub-plots) located in the particular region of interest, whereby only the number of trees in each quadrat is recorded, but not their exact position [11]. If, in addition to counting, the DBH of trees is measured simultaneously, then the count analysis can be extended and divided into proper DBH categories.

Obviously, the most explicit information about tree spatial patterns can be obtained from studies where the total recording of tree positions along with their DBH is performed. However, such measurements are expensive and time-consuming, and thus researchers usually apply certain sampling procedures over the area of interest. For instance, Stamatellos and Panourgias [15] applied random sampling to detect tree spatial patterns across a large forest complex in Greece. While random sampling is undoubtedly an appropriate way to carry out sampling in larger forest areas, systematic sampling might be a better choice when we focus at the stand level. This is because the Poisson model requires independence between sampling plots, which is assured by systematic sampling, whereas in the case of random sampling within a stand, some sample plots may randomly fall too close to each other, or partly overlap. The latter case violates the assumption of independent sampling units that is required for the proper application of the Poisson approach, and in this case, systematic sampling is preferred [14].

Although scattered sampling plots cannot provide explicit distance-related results, the information we may obtain is valuable with regard to species distribution patterns at the global (e.g., stand) level. At this spatial scale, forest ecologists are often interested in the distribution patterns of trees in different DBH categories. Therefore, if the data from superimposed plots across the study area include both tree count and tree DBH, the sound approach is to create wider DBH categories that contain a sufficient number of individuals in each category for the proper application of, e.g., the chi-square test, which is frequently used to test whether the species occurrence pattern is random, i.e., if it follows a Poisson distribution.

Given that they are unaffected by cutting operations, old-growth forests are valuable for studying the real nature of tree species distribution patterns. Various aspects of mixed old-growth forests in Europe have been studied: DBH distributions [16–18], changes in species composition [19], regeneration [20,21], deadwood [22,23], stand dynamics [24,25], and gap dynamics [26–28]. However, the number of studies related to tree distribution patterns in these forests is rather low [29].

The present study was conducted in two mixed old-growth forests in the Dinaric Mountains in order to investigate tree distribution patterns based on discrete (count) data of constituent tree species in the understory and canopy layers at the stand level. Considering the findings from rare previous studies on this topic in beech-coniferous old-growth forests (e.g., [4]), the null hypothesis in this study was formulated. Under the null hypothesis the clumped pattern is expected to be found in the understory and the random pattern in the canopy layer. The alternative hypothesis assumes that the most of the examined tree species, and all tree species combined, will deviate significantly from the clumped pattern in the understory, and likewise, from the random pattern in the canopy layer.

### **2. Materials and Methods**
