Spatial structure parameters based on relationships between nearest-neighbor tree groups

Uniform angle index (W)

Hui & Gadow (2002) described the spatial distribution patterns of a population of trees based on the angles of the vectors joining a reference tree to its n nearest neighbors. One can count only the number of observed angles between two neighboring vectors, which are smaller than the standard angle (α₀), and then divide by the total number of angles, as follows:

When the value of W_iapproaches zero, neighboring trees tend to be distributed more regularly around the reference tree. Conversely, when W_i approaches one, the distribution pattern tends to be clumped. In past studies, the standard angle, describing a situation where all n neighboring trees are distributed regularly around the reference tree, has been considered to be . However, Hui & Gadow (2002) proposed that this situation is rare in nature and not all angles would larger than in an n-tree structure unit. Therefore, the standard angle should be smaller than this value. In this study, we propose that the standard angle should be , consistent with Hui et al. (2007) and Pommerening & Stoyan (2006).

The range of W under a condition of complete spatial randomness (CSR) is the principle for the assessment of distribution patterns of trees in a forest. The value of W under CSR is larger than that seen with regular patterns, and smaller than that seen with aggregated patterns. Consequently, one can easily assess spatial patterns when the thresholds of W under CSR and W of investigated stands are known. Similar to the method that Hui & Gadow (2003) used to identify the range of W_CSR, we generated 1000 stands with random distribution patterns and calculated the average value and standard deviation (σ) of W_CSR; thus, the thresholds of W_CSR is .

Species mingling

Mingling (M_i) is used to express the segregation of species in multi-species forests and is defined as the proportion of n nearest neighbors that are of different species than the reference tree (Gadow et al., 2012). It is defined as follows:

To determine the mingling variable for a whole stand, we summed all M_i values and divided by the number of trees. The bigger the M_i, the more tree species are intermingled.

Diameter dominance

Dominance (U) was defined as the proportion of the n nearest neighbors that were smaller than the reference tree:

Large values of U_iindicate that a given reference tree is suppressed by large trees; for trees in the largest diameter size class we obtained values of U_iapproaching or equal to zero. Generally, U_sp is used to reflect the dominance of a particular species and U_dbhdenotes the dominance of a specific diameter size class.

Edge effects

The treatment of edge trees, those close to plot boundaries, can affect the estimation of indices that include neighbor effects because it is possible that the true nearest neighbors are just outside the study area (Pommerening & Stoyan, 2006). Therefore, we used a buffer zone as an edge-correction method in this study. Trees located in the buffer area were considered neighboring, but not reference, trees. The width of the buffer zone (e.g., d = 5 m) should be neither too small nor too large to avoid wasting tree data, and a larger buffer zone width is required as the number of neighboring trees (two to eight) increases. We used (n + 1 m) to determine the buffer zone width.

Generating stands with different types of spatial structures

The simulated stands patterns and characteristics were based on the existing forest types in multi-species uneven-aged forests of Northern China. Most of the nature or semi-nature forests in this region were protected by the state with seldom commercial harvests. This region has a typical monsoon climate with dry, windy springs and warm, wet summers. Compared to temperate forests in other parts of the world, the tree vegetation in this region shows high species diversity and complex spatial structure. The average tree density in these nature forests is approximately 1,000 trees per hectare.

Different spatial patterns of tree locations

To test the ability of the uniform angle index to assess the distribution patterns caused by different structure-unit sizes, we simulated 20 stands whose spatial patterns approached random distributions with the Winkelmass software (Hui et al., 2007; Petritan et al., 2012); 10 stands were formed with regular patterns and 10 were formed with aggregated patterns. Regular patterns were generated by setting the uniform variation to 80%, which is slightly more regular than complete spatial randomness. The aggregated patterns were generated by setting 500 cluster numbers in 1-ha stands with 1000 trees. The aggregated patterns were slightly more clustered than complete spatial randomness. The pair correlation function, g(r), is considered the most informative second-order summary characteristic (Illian et al., 2008). We used pair correlation function for the patterns of generated stands as a criterion of assessment of spatial distribution patterns at certain scales (Figure 1). There was some regularity (Figure 1a) or clustering (Figure 1b), compared to complete spatial randomness, at 2–6 m in the two different patterns.

Different spatial interspersion of tree species

We simulated three types of species spatial interspersion patterns 1000 times in an area with density of 1,000 trees per hectare, and calculated the average mingling index: (1) complete spatial randomness pattern for tree locations and 10 randomly assigned species; (2) random distribution of tree locations in a stand and the trees located in the buffer zone were assigned to species 10. Then we divided the core area (60 × 60 m) into nine evenly sized cells (20 × 20 m), and assigned each cell one of the remaining species numbered one to nine; thus, trees of the same species located in the core area were aggregated in a cell; (3) random distribution of tree locations, and trees of the same species were mutually exclusive (the distance between two trees of the same species was at least 3 m).

Different spatial interspersion of tree size

Tree locations were distributed randomly within the stand as a whole. We generated trees with diameters of 0–45 cm and evenly divided these into nine size classes that contained approximately the same numbers of trees. Then we simulated four types of spatial-interspersion patterns for the diameter classes. A summary of this process follows.

1. All generated diameters were randomly assigned to tree locations.
2. Trees with diameters of 0–5 cm (smallest class) and 40–45 cm (largest class) were respectively placed in the central cell of the plot, and trees in all other size classes were randomly distributed in other cells; thus, each of the nine size classes occupied a 20 × 20-m cell.
3. Trees with diameters of 0–5 cm were mutually exclusive and all other trees were randomly distributed in the plot. The distance between an arbitrary two trees of the smallest size class was at least 3 m.
4. Trees in the middle size class (20–25 cm) in the central cell of the plot, and the remaining classes were distributed according to the following three spatial interspersion types: (a) smaller classes (0-20 cm) were distributed in the four cells nearest to the central cell, and larger classes (25-45 cm) were distributed in the four cells farthest from the central cell (Figure 2 left), (b) larger classes were distributed in the four cells nearest to the central cell, and smaller classes were distributed in the four cells farthest from the central cell (Figure 2 middle), and (c) each of the smaller and larger classes were distributed in the four cells nearest to the central cell, so smaller and larger trees were equally distributed around the central cell (Figure 2 right). This was performed 1000 times for each of the simulated spatial interspersion patterns of tree size.

The influence of different structure-unit sizes on the uniform angle index

We calculated the uniform angle indices of the 1000 generated stands under CSR, and found that varied with an increasing number of neighbors and that the overall trend had a decreasing-increasing shape (Figure 3). reached its maximum, 0.667, in the two-tree structure unit and it approached 0.5 when three or four neighbors were selected. When the number of neighbors was five to eight, increased with an increasing number of neighboring trees. CSR reached 0.562 in the case of n = 8. The standard deviation (σ) of also varied with n. When fewer neighboring trees were selected, i.e., two or three, σ was very large, being 0.018 and 0.011 for n = 2 and n = 3, respectively. However, it remained stable at ~0.007, when four or more neighboring trees were selected. The threshold of in different structure units is shown in Table 1, determined using ± 3σ.

Table 1. Means of uniform angle indices under complete spatial randomness (W_CSR), standard deviation (σ), and the thresholds of W_CSR for different sizes of structure unit.

Because σ was very large for n = 2 and n = 3, we may have misjudged non-random distribution patterns that were very close to random. Therefore, we tested whether it was possible to assess these distribution patterns. Pair correlation functions for the generated stands in Section 2.3.1 showed that these were non-random patterns at small scales. We recorded the number of stands that were misjudged with respect to distribution patterns in 10 regular and 10 aggregated spatial patterns. For the 10 stands with regular patterns, 8 and 3 stands were misjudged for n = 2 and n = 3, respectively, and no errors were found for other values of n. For stands with aggregated patterns, all stands were assessed incorrectly for the two-tree structure unit, but we detected no errors for structure units with more than two neighboring trees.

Figure 4 illustrates the probability distribution of uniform angle indices under CSR for three and eight neighboring trees (spatial patterns were generated 1000 times). The probability distribution of eight neighboring trees is more detailed than that of three because there are more probabilities of uniform angle indices for the former. The probabilities of W = 0 or W =1 are very close to zero for n = 8, as there are few situations in which all angles are smaller or larger than the standard angle when many angles exist in a structure unit.

Influence of different structure unit sizes on mingling and dominance

When species were distributed randomly within the stand, we found no relationship between the mingling index and the size of the structure unit (Figure 5a). The mingling value remained stable at 0.9 regardless of the number of neighboring trees. When individuals of the same species were aggregated, values of mingling were less than those of random patterns and increased with an increasing number of neighboring trees (Figure 5b). We found a high linear correlation between n and mingling, where the coefficient of determination was 0.992. The lowest value of mingling for n = 2 was 0.11 and the largest for n = 8 was 0.19, the difference between them being 0.08. When the same species were mutually exclusive, namely, when trees of the same species were distributed regularly within the stand, mingling values were larger than those of random patterns and decreased with an increasing number of neighboring trees (Figure 5c), contrary to the trend for aggregated patterns. The difference between the largest and smallest values was 0.05.

Similar to random patterns of species, values of dominance remained stable for the three diameter classes when tree diameters were randomly interspersed within the stand. Dominance values were 0.95, 0.5, and 0.05 for the smallest, middle, and largest classes, respectively (Figure 6a, d, f). The three diameter classes occurred as suppressed, intermediate, and dominant trees.

When trees with diameters of 0–5 cm were aggregated, dominance values were smaller than those of random patterns and increased with an increasing number of neighboring trees (Figure 6b); thus, the pressure from neighboring trees increased accordingly. The difference between the largest value for n = 8 and the smallest value for n = 2 was 0.04. Conversely, when trees of the smallest diameter class were mutually exclusive or were distributed regularly within the stand, dominance values decreased slowly with an increasing number of neighboring trees (Figure 6c).

When trees of the largest diameter class (40–45 cm) were aggregated, dominance values were smaller than those of random patterns and decreased from 0.447 to 0.404 with an increasing number of neighboring trees (Figure 6e); thus, the pressure from neighboring trees decreased accordingly. The difference between the largest value for n = 2 and the smallest value for n = 8 was 0.05.

We observed three spatial interspersion types when trees with diameters in the middle class were aggregated in the core area of the stand (Figure 2). The manner in which dominance varied differed. When trees in the middle diameter class had more small trees around them, their dominance values decreased with increasing n; however, their dominance values increased with n when they were surrounded by more large trees. Dominance values remained stable at ~0.5 when there were equal numbers of small and large trees distributed around trees in the middle diameter class (Figure 6g).

We concluded that changes in mingling values or dominance indices caused by variation in the number of neighboring trees depended on the spatial interspersion patterns of tree species or size.