Competition among individual trees is a fundamental ecological process that plays a major role in population dynamics, survival, growth and species replacement (Peet & Christensen, 1987). By definition, competition is ‘‘an interaction between the individuals, leading to a reduction in the survival, growth and reproduction of the competing individuals” (Begon et al., 1996). Several case studies have been conducted in ecology and forestry to develop, improve or modify different competition indices (CIs). Such indices quantify the competition level for an individual tree and are classified into two major groups of non-spatially explicit indices (e.g. Biging & Dobbertin (1995) and Schröder & Gadow (1999)) and spatially explicit indices (e.g. Hegyi (1974) and Alemdag (1978)). Non-spatial indices are functions of stand level variables, or of the initial dimensions of the trees, and therefore do not require the trees coordinates. Whenever spatial indices are used to measure the influence of local neighbours on a central tree (the subject tree), the dimensions and the relative location of neighbour trees are required for the computation (Tomé & Burkhart, 1989; Corral Rivas et al., 2005).

For several species and forest conditions, the effectiveness of different CIs on tree diameter or basal area growth has been examined (Munro, 1974; Martin & Ek, 1984; Pukkala & Kolström, 1987; Holmes & Reed, 1991; Contreras et al., 2011). Since several aspects of stand density and neighbour sizes influence the tree growth, non-spatial CIs with simple structures are parsimonious to quantify the competitive status of trees in each stand. On the other hand, as ecology is spatial (Berger & Hildenbrandt, 2000), therefore by increasing the interval distance, the negative interaction of neighbours will decrease and spatial CIs take the explicit description of tree spacing into account. Additionally the identity of neighbouring species is an important factor in the characterization of their competitive effect (Bella, 1971; Zhao et al., 2006). Competition can occur among conspecific individuals, plants of same species, and hetero-specific individuals, plants of different species, termed intraspecific and interspecific competition, respectively. The competition behaviour of different species can be differentiated by using Ellenberg et al.’s system (1991). It is the most widely used indicator species system, which compares the response of different species to edaphic and climatic parameters, such as light, temperature, moisture and nitrogen at a 9-point scale for each.

To investigate the effect of competition on the diameter growth of trees, we focused our study on silver birch (Betula pendula Roth). Silver birch occurs naturally in northern temperate and boreal forests and it is an essential ecological and commercial broadleaved tree species (Hynynen et al., 2010). As a pioneer tree species (Fischer et al., 2002), birch is light demanding, and if it grows as a dominant tree with low competitive effects of neighbours, in a stand with relatively wide spacing, birch maintains its vitality and vigorous growth (Hynynen et al., 2010). In Estonia, birch is the second most abundant tree species in terms of forest cover (31.2%) and the coverage is expanding (Yearbook of Forest, 2013). A few attempts have been made to study birch growth related to the negative interaction of tree competitive status in stands (Jõgiste, 1998; Prévosto et al., 1999; Andreassen & Tomter, 2003; Damgaard & Weiner, 2008; Kaitaniemi & Lintunen, 2010).

The main objective of this study was to investigate the adequacy of different spatial and non-spatial CIs to explain single-tree silver birch diameter growth in Estonia. Further objectives were to find the best competitor selection method for Estonian birch stands and evaluate the differences in competitive ability of different species by employing Ellenberg’s species-specific light indicator values. Specifically, we hypothesized that (i) the CIs contribute to explain diameter increments in silver birch; (ii) spatial indices perform better than non-spatial indices; (iii) selecting the potential competitors based on the concept of the influence zone is superior to variable competition zone radii; and (iv) considering the species-specific competition improves the ability of spatial indices to account for growth variability.

As well as the mathematical formulation, the value of a competition index depends on the method used to define competitors for the subject tree (Bigging & Dobbertin, 1992). Among different proposed methods to choose the potential competitors, we tested four approaches. The first two approaches, approaches 1 and 2, were based on the concept of an influence-zone that assumes an imaginary circle whose centre is constituted by the subject tree (Staebler, 1951) and trees inside this circle are competitors. The last two approaches, approaches 3 and 4, identified competitors based on variable competition zone radii, often weighted by dimensions of the subject tree and its neighbours (Daniels, 1976; Ford & Diggle, 1981):

In all the above-mentioned methods, in order to avoid the interference from the competitive effects of non-measured trees beyond the plot borders, we computed CIs only for interior trees on each plot where the neighbours’ information was available for them. After determining the competitors, we calculated the spatially explicit CIs for each subject tree. Four spatially explicit CIs (Sl, SOr, SOdr, and Sdrl1) were based on the influence zone concept and only the first two approaches of competitor selection (CZR_0.4h and CZR_k) were applicable to quantify the mentioned indices. Moreover, the allometric crown radius model, (developed by Lang et al., 2007) was used to calculate the crown radius that was required for quantifying indices SOr and SOdr as well as fitting the Eqs. (7) and (8):

In Table 5 the Spearman’s rank correlation coefficients between the tree diameter growth and non-spatial CIs and also the combination of spatially explicit CIs and the competitor selection methods are presented. Table 5 shows that the competitor selecting approaches significantly affect the growth prediction ability of spatial indices and the competition selection methods of CZR_k3, CZR_0.4h and SCH₆₀ demonstrated greater values of Spearman’s rho, respectively. Among different spatial indices in these three neighbours selecting methods, SdrAng and Heg were well correlated with diameter increment. However, the values of rho for Heg were slightly lower than SdrAng. None of the alternatives of Bitterlich method (BAF₁, BAF₂ and BAF₄) showed to be an appropriate selection method of competitors. Furthermore, BAL and BALMOD as the best non-spatial CIs did not perform better than the superior spatial indices SdrAng and Heg. The results presented in Table 5 are based on the analyses of 2,742 subject trees that were presented in different neighbours’ selection methods and 18 different non-spatial and spatial CIs are quantified and available for them.

The comparison of the linear mixed effect models and the linear multiple models detected the improvement in linear mixed effect regressions in terms of AIC, but the ANOVA comparison between the mixed effect models, with and without an auto-regressive structure, did not show significant remaining within-group correlation (P-value>0.05). Subsequently, the growth model was fit into linear mixed effect regression (Eq. (9)) with no auto-regressive structure for further analyses in this study. All explanatory variables used for Eq. (7) were considered as mixed effects and proved significant (P-value<0.05), also VIF indicated no problem with multiclollinearity, all values being less than eight. In order to test the contribution of selected CIs, they were devised into the recent growth model.

Table 6 illustrates the statistical measures of mixed effect models, including R², RMSE, MSER, AIC, AIC_w, and also DAIC. The indices comparisons were done for different sample size of subject trees based on each neighbour selection method. Generally, the contributions of CIs were significant but not very large in magnitude, and among the CIs added to the model, SdrAng presented the most significant contribution, no matter which competitor selection method was used. After that, Heg was found to be important in efficiency to improve the growth model. Non-spatial CIs of BAL and BALMOD showed less contributions to the growth model than the spatial indices, except for SCH₆₀ where BAL appeared slightly better than Heg CI.

The results of analyses for different age groups (Table 7) demonstrated that competition had stronger prediction ability in younger stands, and spatially CIs proved to be better than non-spatial ones. As shown in Tables (6) and (7), the Ellenberg’s light values performed a slight improvement for some models in order to describe the species-specific effect. The profiles of the R² and AIC did not show considerable variation between the two methods of calculating selected spatial CIs, with and without Ellenberg’s values (Table 6.) However, statistical measures for young stands viewed a slight improvement for including species-specific values in competition quantifications (Table 7).

Computing the correlation coefficient of tree growth, and determining the efficiency of CIs when added to a tree growth model, have been widely used (Burkhart & Tomé, 2012). In the current study, adding the CIs to the growth model slightly improved the model, which can be partially due to the inclusion of relative dimensions of the trees in model. Relative dimensions measure the hierarchical position of the subject tree within the stand, and indirectly indicate the competitive status of the trees (Burkhart & Tomé, 2012).

Results from comparing different CIs proposed that spatial CIs of SdrAng and Heg were the best indices suitable to quantify the competition status of birch trees, respectively. Several studies (Castagneri et al., 2008; Contreras et al., 2011) have reported that SdrAng can describe a greater proportion of the investigated variation in growth models. Also, Heg demonstrated superior performance to non-spatial CIs in many studies (Alemdag, 1978; Pukkala & Kolström, 1987; Holmes & Reed, 1991; Mailly et al., 2003). The indices of SdrAng and Heg assign greater weight to the closer and bigger competitor trees (Wimberly & Bare, 1996) and it was following along the Cole & Lorimer (1994) hypothesis that noticeable competitive stress occurs by immediate competitors surrounding the subject tree crown.

The results we obtained for non-spatial CIs showed that BAL and BALMOD improved the predictive ability of Eq. (9), although in a smaller amount than when using the CIs of SdrAng or Heg. Some studies including BAL or BALMOD found an improvement (large or modest) in model performance (e.g. Biging & Dobbertin, 1995; Corral Rivas et al., 2005). However, similar to our study, several other studies suggested that spatial measures provided more precise growth prediction (Boivin et al., 2010; Contreras et al., 2011), and to the contrary, many studies did not report any superiority of spatial indices to non-spatial ones (Soares & Tomé, 1999; Stadt et al., 2007; Roberts & Harrington, 2008). The superiority of size-ratio CIs of SdrAng and Heg that used the d as indicator of size was probably because of the actual correlation between the subject tree’s diameter increment and its d (Holmes & Reed, 1991); however, the strength of competitive stress explained by such correlations might be unclear (Brand & Magnussen, 1988; Larocque, 2002).

While non-spatial CIs are simple functions of the stand or a tree’s dimensions, the selection of the neighbours that affect the growth of a subject tree is of crucial importance when calculating spatial indices. Concerning the competitor selection methods, the best results were acquired with those based on the influence zone and competition elimination angle concepts. Several studies showed that the competition status of a tree could potentially vary depending on the radius of influence zone (Pukkala & Kolström, 1987; He & Duncan, 2000; Nanami et al., 2005). The CZR _k was a multiple of average distance between the trees in the plot and highly affected by stand density. In our study plots, considering k equal to three and CEA equal to 30° proved to be a good fit to select the adequate number of active competitors. Although some studies (e.g. Alvarez et al. 2003) found better results using a different angle gauge, the angle gauge of 30° provided satisfactory results in some other studies (e.g. Lee & Gadow, 1997; Corral Rivas et al., 2005; Zhang et al., 2009).

The next superior competition selection approach, CZR_0.4h, was simple in practice and in accordance with studies showing that, the optimal influence zone radius strongly depended on the tree’s initial dimensions (D’Amato & Puettmann, 2004; Sims et al., 2009). Considering the third suitable competitor selection method, similar to several other studies, the opening angle of 50°-60° performed well (Biging & Dobbertin, 1995; Pretzsch, 2009; Oheimb et al., 2011) where bigger angles (80° and 100° in this study) mainly decreased the merit of the search-cone method used to detect the competitors (Richards et al., 2008). In contrast to the CZR_k3, the two methods of CZR_0.4h and SCH₆₀ gave more weight to tree height than distance, and since in our study, there was a lack of height measures for all trees, the selecting system of CZR_k3 was preferable to identify competitors for central trees.

Despite the fact that the identity of neighbouring species is an important factor in the characterization of their competitive effects (Bella, 1971; Zhao et al., 2006; Kaitaniemi & Lintunen, 2010; Bošelá et al., 2013), no significant improvement appeared in recalculating the selected indices using Ellenberg’s light values except for young trees. One possible explanation is that in our study plots about two-thirds of the trees analysed were birch with the same light factors. Consequently, giving weight to different species did not significantly change the values of measured competition. In addition, interspecific competition mainly caused by Norway spruce appeared inferior due to differences in temporal growth patterns and shade-tolerance (Tahvanainen & Forss, 2008; Hynynen et al., 2011). Furthermore, the influence of competition on diameter growth was not strongly impacted by the number of species in the local neighbourhood as suggested by Oheimb et al. (2011). The slight improvement in young stands might be due to the nature of Ellenberg’s light values that refer to the preferences of the early stage of the tree life cycle. During the early stage, when light-demanding birch trees rapidly occupy regeneration areas, Norway spruce tends to appear more shade-tolerant; consequently, weighting them differently in competition measures is justified. Moreover the performance of CIs changed slightly by stand development. In young stands spatial indices performed better while in older stands, non-spatial indices showed superior results. In the early stage, pioneer birches grow quite fast and vigorously (Hynynen et al., 2011), and spatial CIs explain competition effects better in dense young stands, since they account for the short distances between the neighbouring trees, that are competing for resources. In older stands, due to mortality induced by different factors, including competition (Sims et al., 2009), the number of trees decline and non-spatial CIs are adequate for competition studies.

The overall results of this study provided a better understanding of competition in birch stands. Although spatial CIs performed better than non-spatial CIs, the reported differences between the spatial and non-spatial indices are relatively small. The spatial indices require tree attributes and locations, and the recording of such information is expensive and time consuming. Therefore, we suggest applying the spatial indices only when studying the competition in the natural development of young stands, where the stands are usually dense, because these types of indices give more weight to trees that are closer to the subject tree. In the middle-aged and old managed stands, an efficient measure of competition is possible by employing the non-spatial indices that do not require as many field measurements.