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## Simple indices

The standard deviation (Husch et al., 2003) of tree dbh measures the variability in tree dimension, thus is a straightforward attribute to calculate. It has been compared in its usefulness to describe stand structure to more complex attributes and indices (Neumann & Starlinger, 2001).

The coefficient of variation of tree dbh measures size inequality (Hutchings, 1997) – it is the percentage of the arithmetic mean of the standard deviation and is used to compare population samples in relative variability.

Basal area is a useful measure of stand density and competition since it incorporates the number of trees and diameters in a stand. It is calculated by summing the section at 1.30m of each individual tree (Husch et al., 2003).

The stand density index (Reineke, 1933) describes stand density by calculating the quadratic mean diameter and number of stems per unit area. It uses an index tree diameter of 25cm in Europe (10 inches in the U.S.) and a constant b of -1.605, which Reineke (1933) found was consistent with several species and age and site quality independent (Pretzsch & Biber, 2005).

**Adapted species biodiversity indices**

A straightforward way to describe forest complexity would be to just determine the number of size classes in a stand (richness) (Valbuena et al., 2012). The following six indices are based on size classes.

Adapted from the Shannon diversity index for species (Shannon & Weaver, 1949), the Shannon index for diameter classes describes the class richness and relative abundance of every class as well as its rarity by measuring the uncertainty of a tree belonging to a class as a weighted average (Buongiorno et al., 1994, LeMay & Staudhammer, 2005, Pommerening, 2002, Staudhammer, 1999, Valbuena et al., 2012). In this case, the Shannon index has a minimum of zero when all trees are in the same diameter class and a maximum of the logarithm of the total number of classes when trees are evenly distributed in all diameter classes (Lexerød & Eid, 2006).

The Simpson concentration index (Simpson, 1949) for diameter classes is an index of dominance (weighted to the most abundant class) and expresses the chance that any two random trees belong to the same size class (Lexerød & Eid, 2006, Valbuena et al., 2012). A reciprocal form of the index was used in this study (1/index value), also known as the inverse Simpson index of diversity, so that the index values increase with any diversity increase (Lexerød & Eid, 2006). The Simpson index ranges from a minimum of zero to a maximum equaling the total number of diameter classes. It accounts for dominance in a similar way to Shannon (the more dominant a class, the higher its Simpson index).

Described as one of the “most satisfactory indices of diversity available” (Magurran, 2004), the Berger-Parker index (Berger & Parker, 1970) is another dominance measure independent of the number of classes of diameter. It is the reciprocal of probability of the most dominant diameter class, therefore ignoring the frequency of the rarer size classes (Lexerød & Eid, 2006, Valbuena et al., 2012). The reciprocal value of the Berger-Parker index ranges from 0 to 1 and increases with more evenness and lower dominance.

**Adapted equitability biodiversity indices**

The Shannon or Pielou index (Pielou, 1969) is a measure of evenness between size classes as it is the ratio of absolute diversity to the maximum diversity of classes possible (Lexerød & Eid, 2006, Staudhammer, 1999).

The Simpson evenness (Magurran, 2004, Pielou, 1969) measures the relative abundance of the size classes in a stand – and therefore normalizes the figures with respect to the maximum value possible (Valbuena et al., 2012).

Both Shannon evenness and Simpson evenness indices range from 0 (distribution between size classes not even) to 1 (distribution between size classes completely even) and measure the abundance of the different dbh classes within a forest stand.

**dispersion indices**

The Gini index (Gini, 1921) measures size inequality and quantifies the deviation from perfect equality by calculating the area under the Lorenz curve derived by plotting the cumulative size classes of trees per hectare against the proportions of the number of stems per hectare (Bilek et al., 2011, McCarthy & Weetman, 2007). It has been described as an indicator that performs better than other forest structure indices at gauging the heterogeneity of tree sizes (Lexerød & Eid, 2006, Valbuena et al., 2012). The Gini coefficient has a value close to 0 for a homogeneous forest stand and closer to 1 for a heterogeneous forest stand.

**Descriptors of histogram’s shape**

Skewness or asymmetry of the diameter distribution is an important diversity measure and is described as the departure from symmetry, with the assumption of normal distribution (Hui & Pommerening, 2014, McCarthy & Weetman, 2007, Siipilehto, 2011, Sterba & Zingg, 2006). Lorimer’s index of symmetry (Lorimer & Krug, 1983) can distinguish between descending monotonic, skewed and symmetric unimodal tree diameters (McCarthy & Weetman, 2007). The skewness and Lorimer index are both related to the shape of the histogram and its symmetry, whereby a symmetric distribution would have a skewness of 0 and a lorimer index of 0.5. A positively skewed distribution would have a positive skewness (>0) and a lorimer index above 0 and under 0.5 while a negative skewed distribution would have a negative skewness value (<0) and a lorimer index above 0.5 and under 1. The main difference between the two is that Lorimer takes into account the difference between median and mode while for skewness the distribution is symmetrical around the mean and median.

Kurtosis is an indicator of the diameter distribution as well that describes the shape (or peakedness) of a distribution that is assumed to be normal; it tends to be more influenced by a few extreme differences from the mean than a lot of small differences (McCarthy & Weetman, 2007, Siipilehto, 2011). The kurtosis distribution analysis distinguishes between the normal distribution or mesokurtic (=3), platykurtic (<3) and leptokurtic (>3) distribution. A higher kurtosis means there is more variability due to a few extreme differences from the mean.

**Distance-dependent indices**

Spatially explicit indices indicate whether a spatial point process has complete spatial randomness (CSR), i.e. if it follows a non-heterogeneous Poisson process, regularity or clustering in a stand (Diggle, 1983). The distance-dependent indices are the following:

### Nearest neighbor indices

An angle-based index, the mean directional index (Corral-Rivas, 2006) is a nearest neighbor tree parameter index representing the spatial arrangement of trees. It aims to characterize a reference tree by the different directions under which the n nearest neighbors can be seen (Corral-Rivas, 2006, Motz et al., 2010). It is easy to calculate and interpret as it does not require tree to tree distances and is a good estimation of non-randomness: if it has a value of 0, it means the spatial arrangement is in a square lattice and increasing values indicate more clustered patterns (Corral-Rivas et al., 2010, Motz et al., 2010, Szmyt, 2014).

#### Distance-dependent indices and combination indices

Since most spatially explicit indices are more accurately estimated by using fixed angle stands instead of angle count sampling (Motz et al., 2010, Zenner, 2014), spatial indices have been calculated using a 10 m radius from the center of the plots (includes all the trees above 7.5 cm). Indices were also calculated at 20 m and 40 m (includes trees identified by angle count) for comparison purposes, however unless a significant result was identified, only the 10 m results will be used for accuracy purposes. Since spatial indices require that at least two trees be present in a stand, stands with less than two trees were removed from the data, leaving a total of 244 stands for the calculation of the mean directional index, the Clark Evans index, the size differentiation and the size dominance index. The Hopkins Skellam index and the L function index required a minimum of one tree per stand and therefore included a total of 249 stands at 10 meters.

Bearing (azimuth) coordinates for each live tree from the center of the plot was converted to xy coordinates in order to be able to calculate the spatially-implicit indices. Each azimuth in grades was converted to an arithmetic angle by converting the grades to degrees (degrees=grades*360/400). The distance and bearing was then translated into a Cartesian coordinate:

x coordinate = distance * cos(bearing degree).

y coordinate = distance * sin (bearing degree).

Each coordinate was then included in a plot with maximum and minimum coordinates for x and y from the plot center. In order to correct for the edge effect, calculations of the Clark-Evans index and the Hopkins-Skellam index took place within a finite study area with a radius of 10 m, 20 m or 40 m – points outside this boundary were not included in any calculations. The L function also took place within a finite window area and was calculated for five default distances from 0 to 10 meters – the L function single index value for 5 m and 10 m was then selected for each modality.

The mean directional index, size differentiation and size dominance indices are tree-parameter indices, nevertheless their arithmetic mean can be calculated to describe the structure of a stand (Motz et al., 2010, Pommerening, 2006). These three indices were calculated for the four nearest neighbors (n=4). Since the number of trees for each stand at a 10 meter radius is under 100, no edge correction was used since in order to avoid large bias values as explained in Pommerening and Stoyan (2006). Nonetheless, no edge correction results were compared with the nearest neighbor edge-correction concept (NN1) results (which eliminates trees not within the plot), which is proposed by Pommerening and Stoyan (2006) particularly for angle-dependent indices. The differences detected between the two sets of results were minimal for the mean directional index, size differentiation index and size dominance index, therefore no edge correction results are used here.

**Table of contents :**

**1. INTRODUCTION **

**2. MATERIAL AND METHODS**

2.1. Description of the study sites

2.2. Data collection

2.3. Description of the indices

2.3.1. Distance-independent indices

2.3.2. Distance-dependent indices

2.3.3. Combination indices

2.4. Calculation of the indices

2.4.1. Distance-independent indices

2.4.2. Distance-dependent indices and combination indices

2.5. Data analysis

**3. RESULTS **

3.1.1. Distance-independent indices

3.1.2. Distance-dependent indices

3.1.3. Combination indices

3.1.4. Correlations between indices

**4. DISCUSSION **

4.1. Characteristics of managed and unmanaged stands according to the distance-independent indices

4.2. Differences between lowland and mountain sites

4.3. Indices that rejected the null hypothesis

4.4. Spatial indices, Plot size and Angle count sampling method

**5. CONCLUSION **

**6. REFERENCES**