**3. Results**

The values of the *Ic* index and best fitting models presented in Table 1 and Figures 1 and 2, suggest that in the two old-growth forests, the same distribution pattern, but in some cases different distribution patterns, may characterize a tree species. For instance, understory beech trees (≤27.5 cm DBH) had a clumped pattern in both of the studied old-growth forests. However, in the canopy layer (>27.5 cm DBH) this species exhibited a random pattern in Janj and a regular pattern in Lom. Silver fir trees in both the understory and canopy layer were characterized by a random pattern in both old-growth forests. Similarly, Norway spruce generally exhibited a random pattern, except for its understory trees in Lom, which exhibited a clumped pattern. However, when both conifers were jointly analyzed (fir and spruce as one variable), their distribution in the understory of both old-growth forests was clumped. On the other hand, the joint distribution of conifers in the canopy layer remained random as in the case of single coniferous species. When all trees (all species combined including beech, fir, spruce) were considered, they clearly exhibited a clumped pattern in the understory, while their distribution in the canopy layer varied from random to regular in Janj and Lom, respectively (Table 1, Figures 1 and 2).

**Table 1.** The patterns of tree count data in the two studied old-growth forests. The values of the dispersion index (*Ic*) are provided in brackets.


In the understory layer in both old-growth forests, *Ic* index values ranged from 1.06 to 5.45, whereas these values for the canopy layer varied from 0.42 to 1.39. Generally, the Poisson distribution was the best fit to model species count data when the respective *Ic* index values were between 0.68 and 1.42. For index values below 0.68 and above 1.42, the binomial distribution and negative binomial distribution were found to be the best fitting models, respectively.

With respect to the count distributions of understory trees (Figure 1), the span of the beech counts in plots was much greater compared to that of fir and spruce, while the conifers combined resembled the beech distribution in the Lom old-growth forest. In this stand layer, the negative binomial distribution was the best fit for beech counts, for conifers combined, and for all trees combined in both old-growth forests. The only inconsistency was for spruce trees as the counts for this species in the understory were best modeled with the Poisson distribution in Janj (Figure 1c), while in Lom the negative binomial distribution was the best fit for this species (Figure 1h). Fir understory counts were best fitted with Poisson distributions in both studied old-growth forests. What was also interesting with respect to the understory figures, was that all plots contained beech trees, while the absence (plots with 0 tree count) of fir and spruce ranged from 8 to 21 (20% to 52.5% of plots), respectively.

**Figure 1.** The observed and expected tree counts for individual tree species, for conifers combined, and for all species combined in the understory layer (7.5–27.5 cm diameter at breast height (DBH)) in the Janj (left: **a**–**e**) and Lom (right: **f**–**j**) old-growth forests. The expected counts were shown based on the models that best fitted the observed counts: negative binomial distribution (NBD) and Poisson distribution.

Contrary to the understory layer, the span of count distributions in the canopy was fairly similar for beech, fir, and spruce. Also, in contrast to the understory layer where the negative binomial model prevailed, the trees in the canopy layer followed a Poisson distribution in most cases (Figure 2). In this stand layer, the binomial distribution was found to be the best fitting model only for beech counts and for all trees combined in the Lom old-growth forest (Figure 2f,j, respectively). It is important to emphasize that all fitted models were significant at the 0.05 α-level, except in the case of all trees in Lom (Figure 2j), namely, in the latter case, a Poisson and negative binomial distribution clearly deviated from the observed counts, and a binomial distribution followed it much better. Therefore, the binomial distribution was selected as the best fit. However, it should be noted that fitting all canopy trees in Lom, even with the binomial distribution was also non-significant. Consequently, explaining real (observed) tree count distributions with theoretical models seems to be more challenging in the case of a regular data pattern than in the case of random and clumped patterns.

**Figure 2.** The observed and expected tree counts for individual tree species, for conifers combined, and for all species combined in the canopy layer (trees with DBH > 27.5 cm) in the Janj (left: **a**–**e**) and Lom (right: **f**–**j**) old-growth forests. The expected counts were based on the models that best fitted the observed counts of canopy trees: Poisson distribution and binomial distribution (BD).

#### **4. Discussion**

This study showed that in mixed old-growth forests composed primarily of beech, fir, and spruce, the aggregated (clumped) pattern mainly characterized understory beech trees with a DBH between 7.5–27.5 cm. In the canopy layer (>27.5 cm DBH), the count data patterns of beech trees were more variable compared to fir as the counts of the latter in both studied old-growth forests followed a Poisson (random) distribution in the understory as well as in the canopy layer. Spruce clearly exhibited a random pattern in the canopy layer, whereas its count distributions followed a Poisson and negative binomial distribution in the understory of Janj and Lom, respectively. Contrary to single coniferous species, joint conifers (fir plus spruce) had clumped understory patterns that were best modeled with a negative binomial distribution, whereas in the canopy layer, their common pattern was random and followed a Poisson distribution. Interestingly, all trees (all species combined) exhibited patterns identical to those of beech in the understory and canopy. This study partly confirms the results reported by Gu et al. [40], which found that the degree of tree clumping decreases from juvenile to adult stages. In addition, it is important to note that different values of dispersion index for beech and for all trees also indicate different degrees (different intensity) of clumping on one hand, or regularity on the other.

The quadrat count method applied in this study has certain advantages and disadvantages compared to spatially-explicit methods where the distance between trees is used. Therefore, the results of this method should be treated with caution as they partly depend on the size of the sample plots [11]. When the purpose is to compare different studies that applied different sample plot sizes, this issue may be solved by recalculating tree counts to one (equal) sample plot size for all compared sites. However, such an approach is feasible only when the raw data are available or readable from figures; otherwise, comparison of the dispersion index between sites where the size of sample plots is different must be interpreted with caution. The second limitation of the quadrat count method is its inability to detect the spacing distance between trees, which might be useful information when the trees are clumped and/or regularly distributed. Consequently, there is no insight into the scales at which processes such as positive and/or negative autocorrelation between trees occur [9]. For instance, the application of the Ripley K- function and/or *g(r)* pair correlation function [41] not only provides information about tree patterns (random, clumped, or regular), but also information about facilitation (positive interaction between neighboring tree individuals) and competition (negative interaction between trees), which usually occur at different spatial scales within a forest stand.

Nevertheless, when spatially-explicit data are limited or missing, the quadrat count method seems to be a sound analytical approach to investigate whether the point pattern associated with individual trees in the stand exhibits complete spatial randomness or a clumped or regular pattern. This method also answers the question of how densely the sample plots are populated by constituent tree species, thus, the absence of any species of interest from a large percentage of plots may be an indication that something is wrong or that something unusual is happening with that species. Such information cannot be obtained based on traditional DBH distributions.

For instance, classical DBH distributions of a tree species may have virtually the same forms (shapes), when in fact a species may have very different data count patterns. Let us consider two cases: (a) a tree species may be densely present in very few plots, while at the same time it might be missing in most others; and (b) it may be regularly present in a similar number in all, or almost all, plots. The difference between these two cases cannot be detected by classical DBH distributions, and therefore, there is a good reason to supplement them with the quadrat count analysis whenever the goal is to investigate tree species distribution in detail.

#### **5. Conclusions**

Models based on count data are not meant to replace models based on continuous variables (e.g., DBH), but they may complement them by providing additional information about species count distributions across a forest stand. As previous studies in the Janj and Lom old-growth forests [30] have shown, a species distribution based on DBH as a continuous variable may indicate a form of sustainable distribution such as negative exponential or rotated sigmoid. However, such information is only partly useful to forest managers and nature conservationists as conventional DBH distributions do not disclose how a species is spatially distributed over an observed area. On the other hand, the distributions of species count data, as applied in this study, reveal how a tree species is distributed within a forest stand, that is, whether it more or less equally occurs in all parts of a stand, exhibits a random pattern, or tends to group in a few plots. This study also demonstrates that the above information can be obtained separately for trees in the understory and canopy layers providing that both data types (species counts and DBH) are available.

So far, modeling of species count data has usually been performed on single large plots, however, this study shows that it can be effectively conducted on small scattered plots as well. Such an approach might be used to supplement future studies of DBH distributions based on scattered plots, especially in mixed forests. Then, conclusions about sustainability of a tree species would be more reliable. The observations of real species counts and fitted theoretical models are important as they reveal not only the count (abundance) and the variability of an observed species in sample plots across a study area, but they also show the number of empty plots (absence of a species). In a spatial context, specifically at the stand level, such information might be highly useful to forest managers and nature conservationists interested in monitoring and sustaining a species at such a spatial scale.

**Funding:** The research was supported by the project Innovative forest MAnagEment STrategies for a Resilient biOeconomy under climate change and disturbances (I-MAESTRO) under the umbrella of ForestValue ERA-NET Cofund, among others by the National Science Centre in Poland (NCN). ForestValue has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement Nr 773324. The APC was funded by the Ministry of Science and Higher Education of the Republic of Poland for the University of Agriculture in Krakow for 2020.

**Acknowledgments:** The author gratefully acknowledges the support of Ivan Bjelanovi´c (previously of the University of Belgrade, Serbia, now the Department of Natural Resources of the Canadian Government) during the fieldwork in the Janj old-growth forest, and also thanks to the small but great research team from Turin (Italy) led by Renzo Motta that helped with data collection in the Lom old-growth forest.

**Conflicts of Interest:** The author declares no conflict of interest.
