4.2. Effect of Mixing on the Spatial Pattern of Tree Distribution
The spatial distribution of trees in stands is influenced by numerous factors. It is most influenced by the management system, the origin and structure of stands, the composition of the species that build them, developmental stages, natural disturbances, habitat conditions and abiotic conditions to which they have adapted their development. In intensively managed stands, the distribution of trees tends towards a regular (random) distribution [
59,
60]. In stands that are not subjected to management treatments, active selection and differentiation processes influence the regulation of tree distribution with a tendency toward a regular distribution [
40,
41], i.e., random distribution [
60,
61], which was confirmed by this research. These pronounced processes in mixed stands of the studied species affected the random distribution of trees. The regularity of distribution is especially pronounced in trees with larger diameters [
40,
41], which is also the trend observed in this research (SP 7, SP 8).
The tendency towards clumping of trees at lower distances in the stands of Serbian spruce (SP 1, SP 2) can be explained by microhabitat conditions (form of the micro-relief conditioned by the parent rock, a mosaic shallow initial phase of the soil) and the relative absence of other species with different bio-ecological characteristics which would contribute to the process of intensive competition and differentiation. The formed homogeneous groups in the coves and at the crossings between ridges, which intersect the entire locality, condition the clumping of trees at relatively larger distances. The clumping of Macedonian pine trees at lower distances (SP 5, SP 6) can be explained by the specifics of the species development in high mountain climatic conditions. The relations of the spatial distribution of the trees can be divided into two groups; I. with a lower degree of mixing (SP 1, SP 2; SP 5, SP 6), where the trees tend to clump (group) and II. with a higher degree of mixing (SP 3, SP 4; SP 7, SP 8), where the trees tend to be randomly to regularly distributed. Wider distributions of dimension (
Figure 3 and
Figure 4) indicate an increasingly pronounced influence of the separation of ecological niches of different species in mixed stands and the dimensional unevenness and intensive separation of taller and shorter trees into strata. This fact also affected the spatial arrangement of the trees.
4.3. Diversity Indices as an Appropriate Measure of Stand Differences (Td, GC)
The diameter differentiation index
Td based on the researched references shows different values in different stand situations. The lowest values are typical of even-aged stands that are actively managed, 0.11 in taigas [
62], 0.13–0.21 in pine cultures of different ages [
41], 0.21 in an even-aged beech stand [
8], 0.25 in young Douglas fir culture [
8], 0.30 [
59], and 0.34 [
60] in even-aged stands of different forest types. Slightly higher average values of
Td were found in uneven-aged stands of beech 0.27–0.42 [
63], 0.36 [
64], and 0.42 [
8] in mixed stands of beech. In a series of permanent research plots in the Czech Republic [
65,
66,
67,
68] on a similar sample in protected mixed stands of spruce, fir and beech values were determined to be from 0.41 to 0.55 [
67] and in managed mixed stands of beech of different origin they were from 0.36–0.49 [
68]. In relict pure stands of Scots pine lower values (0.20–0.33) were found [
65]. In research on mixed stands of Scots pine without management treatment [
66], a trend of decreasing the degree of diameter differentiation (from 0.37–0.48 to 0.36–0.42) was determined in the analyzed period of 15 years on all permanent research plots. The inventory of different forest types in the Austrian Alps determined the average value of the
Td index in stands of 0.37, and values significantly above this average (0.50) were found in two-story (multi-story) stands [
69]. In the selected stands, the determined mean values of the
Td index are 0.43 [
64] and 0.42 [
59]. The highest
Td values are typical of virgin forests, and they range from 0.47 [
70] to 0.76–0.78 [
60].
By comparing the presented Td indices, the differentiation of diameters has characteristic higher or lower values in relation to the developmental and structural type of stands. These differences indicate, but do not determine, the stand type. In non-managed forests, the differences are the result of competition between species and within the species during development, and in forests that are regularly managed, they are a direct consequence of these relationships and the nature of applied management treatments.
The determined values of the index from 0.24 to 0.33 indicate a simpler type of stand, which only confirms the previous analysis of the height and diameter structure.
In the practical use of the
Td index, it is important to pay attention to the distribution of index values of structural groups, as well as the use of appropriate measures of central tendency. The approximately normal values of the
Td index distribution indicate the possibility of using mean values [
62] as a representative parameter in the analysis of the relationship, which is the case here. For significant deviations from the normal distribution, the use of mode [
62] or median [
69] is adequate.
The Gini coefficient
GC has the lowest values in young even-aged stands of 0.15–0.30 [
71], 0.22 [
60], and 0.25 in two-story stands [
69]. Higher values are typical of mixed and uneven-aged stands of different compositions (e.g., 0.37–0.54 [
19], 0.50–0.58 [
71], 0.49–0.57 [
72], and 0.35–0.52 [
73]). Values above 0.60 [
71] are characteristic of selective stands and virgin forests. The mean values in the selection stands of the Austrian Alps are 0.63 [
59]. The highest values of the index were found in beech and spruce reserves and virgin forests of the southeastern Carpathians 0.69–0.71 [
74], beech and hornbeam in the Czech Republic 0.67–0.75 [
60], beech, fir, spruce in Bosnia and Herzegovina 0.67 [
70], and beech in Serbia 0.45–0.52 [
72]. The values presented in different stands justify the use of
GC as an adequate measure of stand inequality, which also indicates but does not determine the stand type.
In relation to the distribution of the analyzed dimensions, the indices show certain regularities. Stands with a normal distribution have lower
GC values compared to stands with the inverse J-shape [
46,
59] and regular distributions [
46]. In this research, two stands with a statistically significant normal distribution (SP 1, SP 6) have the lowest
GC values. This confirms the assumption that
GC values increase with the deviation of the distribution from the normal shape [
47,
75]. The diameter distribution in uneven-aged stands is reflected in higher and more constant index values compared to even-aged stands [
47]. In the research of different forms of pure and mixed stands of spruce, fir, and beech of the Eastern Carpathians [
75]
GC has been shown to be a useful tool in differentiating different structural types of stands. Regardless of the established justification and superiority in relation to other indices [
46], the possibility of obtaining the same values in different structures (as with the previous index) limits its application [
76]. That shortcoming requires the analysis of the flows of the Lorentz curve on which the index is based or the use of adequate additional parameters in determining the differences of the analyzed stands [
76].
In the practical use of this index, as a useful tool in forest management, it is necessary to pay attention to the impact of the callipering threshold on its value [
75], the type and size of the sample used [
77], as well as the index value in the structures of transitional forms in the process of stand regeneration [
47].
4.5. Correlation of the Investigated Parameters of Structural Diversity
There is not much literature data on the impact of mixing on the aggregation index. Contrary to the obtained results (
Table 6), in stands with active management, the established ratio has a negative character of low intensity [
62].
The pronounced link between the vertical fulfillment of the
Arel profile and the mixing of stands is based on their mathematical correlation. The obtained values confirm the advantage of using the index in stands with different numbers of species [
5], as well as the impact of deviation of structure from pure single-story stands on the growth of index values [
78].
The impact of mixing on dimensional diversity indices is indirectly expressed through the impact on stand structure. Differences in the distributions of the investigated stands with increasing mixing had a positive effect on diameter differentiation indices (
Td,
GC). A study of different empirical and theoretical stand situations [
46] found a strong impact of mixing on
GC values. Significant differences in
GC values were found between the pure and mixed even-aged stands of the most common species of Central Europe [
79]. Mixed stands had on average significantly higher
GC values (0.46) than pure stands (0.36), with an established correlation of medium strength.
The moderately strong positive correlation found in studies of the impact of mixing on the
M and
Td differentiation indices in virgin forests [
70] and actively managed different forest types [
62] was also confirmed by this study.
Prior research in stands of even-aged and almost even-aged structure found a strong correlation between
Td and
GC [
80], which is also confirmed by this study. In contrast, the weak correlation of these indices in stands of more complex structure, virgin forests [
70], diverse stands [
80], and stands of different types [
71], emphasizes the impact of structure on the change in their relationship. The reason for this should be sought in various mathematical assumptions of the above indices.
Since the hyperbolic tangent index
S is a new index, no research has been performed on its correlation with other structural parameters. In addition to the tree characteristics diversity index, it has also been proposed as a useful tool for researching absolute and relative growth rates [
48]. Considering the concept of using the index, the relationship between the
S index and the mixing index
M depends on the distribution of the observed characteristics of the admixed species in stands with different degrees of mixing. In this research, it has a weak positive correlation with the stand mixing index.