Forest Ecology and Tree species mixture e ﬀ ects on stem growth vary with stand density – An analysis based on individual tree responses

Overyielding in mixed species stands is expected to vary with stand density, but only a few studies have quanti ﬁ ed this. We used individual-tree growth data from permanent sample plots on 83 sites in Switzerland representing a three-species mixture between Norway spruce, Silver ﬁ r, and European beech. Basal area growth models for all three species indicated signi ﬁ cant interactions between a competition index ( CIs ) and species composition in the neighborhood. The spatially-explicit CIs indicates stand density within a 10- m radius neighborhood, and the contribution of individual species to the CIs indicates the species composition. Given the rich vertical stand structure in the sample plots, which were often managed by single-tree selection cutting, it is not surprising that interactions between CIs and species proportion varied with relative tree height. We used the individual tree growth models to simulate stand growth with and without species composition e ﬀ ects. A sample plot with its real stand structure in terms of tree size and position was homogenized in terms of species composition at the tree level. By varying the stand density of this plot, we demonstrate how overyielding increases with stand density. In the given mixture, it is mostly beech contributing to the observed overyielding in dense mixed stands. Stand density therefore needs to be considered more explicitly when studying growth and yield of mixed species stands.


Introduction
Productivity of mixed species forests has frequently been reported to differ from that of monospecific stands. However, plenty of evidence has been accumulated showing that these effects vary with the tree species involved, site quality, stand structure, and over time due to ontogenetic development and stand dynamics (Forrester, 2014;Forrester and Bauhus, 2016). Stand density has also frequently been presented as an important effect on productivity in mixed stands (e.g., Forrester, 2014;Forrester and Bauhus, 2016;Pretzsch and Biber, 2016). However, empirical research has often relied on experiments or temporary sample plots established in stands of high density and not analyzed the effects of stand density explicitly. Despite these objectives, stand density in the studied sample plots was frequently below maximum density, affected by thinning, or varied substantially between plots or over time. Despite early warnings that most experimental designs are unable to address the effect of stand density on productivity in mixed stands (Kelty and Cameron, 1995;Vanclay, 2006a), few studies have tried to correct for stand density effects in analyses from those data sources. Individual-tree-based approaches for the analyses of growth records and experimental designs have been proposed (Kelty and Cameron, 1995;Vanclay, 2006a) and also been applied frequently for other objectives (e.g., Canham et al., 2004). However, stand density effects on productivity of mixed stands have not been addressed explicitly with these approaches, except for a few examples presented below. Consequently, we are still lacking a quantitative empirical understanding of these effects, which are essential for managing mixed species forests.
Stand density changes dynamically over time. Young stands are most often established at densities far below maximum density and intraspecific as well as interspecific competition between trees only starts as the canopy closes and intensifies as the density increases. Few stands are managed to develop along the self-thinning line, which would indicate maximum competition and significant interspecific effects on total productivity. Maximum stand density for mixed species stands varies with site quality (Condes et al., 2017;Ducey and Knapp, relative density often contributes to the uncertainty about effects of species mixtures on stand productivity. Both the unknown maximum density and the dynamic changes in stand density make the analysis of density effects in mixed stands an almost intractable problem for experimental research at the stand level. The general objective of our study was to quantify the effects of stand density on stem growth in mixed stands. Given the lack of appropriate experiments controlling for stand density, we used data from long-term experiments in Switzerland that varied in species composition and stand density. Individual-tree-based methods were used to analyze the data. The long time series also allowed us to account for effects of stand dynamics and ontogenetic development on inter-and intraspecific competition. Many sample plots in these experimental series have been managed according to the single-tree selection system and are therefore uneven-aged and contain trees with a large range of dominance. The species mixture that supplied most individual tree records for our analyses was the common three-species mixture in this region of Norway spruce (Picea abies (L.) Karst.; hereafter spruce), silver fir (Abies alba Mill.; hereafter fir), and European beech (Fagus sylvatica L.; hereafter beech). Given the data used to analyze the effects of stand density, the effects of stand structure also need to be considered when formulating hypotheses. Stand structure is here defined as the degree of size inequality of trees in a stand in terms of tree diameter and height.

Effects of stand density
Few studies have addressed the effect of stand density on productivity in mixed stands on the stand level and these have used a range of different methods. Controlled experiments in mixed stands have often not addressed this question, but some examples have been reported. Overyielding in young stands of Douglas-fir and western hemlock only occurred at the highest stand density, which was also characterized by the most distinct stratification between those two species (Amoroso and Turnblom, 2006). Two other conifer mixtures did not show any significant effect of stand density on relative yields in young stands (Garber and Maguire, 2004). However, in the stratified mixture overyielding was observed in the densest stand, but not in more open stands. Mixing increased productivity earlier in denser plots of Eucalyptus globulus and Acacia mearnsii, and this mixing effect remained higher in the more dense mixtures than the less dense as the stands developed Forrester et al., 2004;Khanna, 1997). Following three species in two-species mixtures over time at different densities showed that ontogenetic differences between the species depended on stand density and led to different results in terms of overyielding (Boyden et al., 2009).
National forest inventory data have also been used to describe how mixture effects vary with stand density. For some species, mixing affected standing aboveground carbon stocks only at higher stand density (Woodall et al., 2011). del Rio and Sterba (2009) found no significant effect of density on species proportion effects in their growth models, but presented minor effects in prediction plots based on back-transformed variables. Condes et al. (2013) found opposite effects of density on species proportion effects for Scots pine and beech. Beech is often found in lower canopy positions in this mixture and benefits from increased pine proportions by reducing intraspecific competition, but only at high stand density. Pine on the other hand is not affected by beech admixture for the most common medium to high stand densities, but negatively affected by higher pine proportions at low density. For the spruce-fir mixture (Huber et al., 2014) and the spruce-beech mixture (Houpert et al., 2018) in Switzerland, no interactions were found between species proportion and stand density on stand-level basal area increment.
Using temporary sample plots in stands composed of three different species that have been maintained at low density during early stages of stand development, due to planting density and precommercial thinning, no overyielding in the standing aboveground biomass was found, which was interpreted as a lack of species interaction due to low stand density (Himes and Puettmann, 2020). However, current and historic stand density was insufficiently documented in this study to allow conclusions about the effect of density on species interactions.
At the individual tree level, competition indices have been used to quantify stand density in the local neighborhood and can at the same time be used to quantify intra-and interspecific competition. Individual tree growth models with explicit species effects have been estimated from data in a number of studies (e.g., Canham et al., 2004;Kelty and Cameron, 1995;Kunstler et al., 2012;Papaik and Canham, 2006;Pukkala et al., 1998;Pukkala et al., 1994;Ratcliffe et al., 2015;Riofrio et al., 2019;Riofrio et al., 2017;Vettenranta, 1999;von Oheimb et al., 2011). Using these methods, intraspecific competition is often reported to reduce growth more than interspecific competition (e.g., Coates et al., 2009;del Rio et al., 2014;Forrester et al., 2011;Ngo Bieng et al., 2013;Perot et al., 2010;Perot and Picard, 2012;Zhao et al., 2006). Some studies have demonstrated how species-specific competition changes with tree dominance or ontogeny Coates et al., 2009;D'Amato and Puettmann, 2004;Forrester et al., 2011;Gomez-Aparicio et al., 2011;Manso et al., 2015). Other studies have demonstrated how species-specific competition changes with site quality (Boyden et al., 2005;Coates et al., 2013;Condes and del Rio, 2015;Forrester et al., 2013). Site and climatic influences on the balance between intra-and interspecific competition between spruce, fir, and beech in two-and three-species mixtures was demonstrated with the same method in national forest inventory data from Switzerland (Mina et al., 2018a). We only found two studies that have used tree growth models to analyze the interaction between stand density and mixture effects. For a mixture of spruce and fir, increasing overyielding has been predicted as stand density increases for a simulated stand of uniform trees (Forrester et al., 2013). Using an individual tree-based growth simulator (Bristow et al., 2006) with some simple additional assumptions, Vanclay (2006b) showed an optimum pattern over density for volume growth in mixed stands. Using a similar method, observed stand structures, and mean field theory, the effect of stand structure on mixture effects has been studied (Perot and Picard, 2012). Overyielding in the oak-pine mixture was mostly related to the intraspecific competition and spatial structure of the more shade-tolerant oak species.
Given that only few studies explicitly addressed the effects of density on mixture effects and that the effects varied largely between species and response variables, a common theory cannot be formulated. We hypothesize that mixture effects on stem diameter growth increase with density for the three shade-tolerant species in our study due to increasing inter-and intraspecific competition at higher neighborhood densities.

Effects of stand structure
Size inequality in mixed species stands has been reported to increase as a consequence of interspecific competition, and to affect interspecific competition. Using stand-level structure indices, Danescu et al. (2016) demonstrated that stand structure is important in explaining variation in basal area increment of individual trees and stands in structurally diverse spruce-fir-beech stands. Using individual tree growth models, large effects of stand structure variation on stand-level mixing effects were demonstrated in oak-pine stands (Perot and Picard, 2012). del Rio et al. (2014) demonstrated that competition between beech and three different admixed species depended on the relative size of the competitors using plot-level classification into larger or smaller trees. For the beech-fir mixture, the density of larger competitors explained variation in basal area growth best for both species. Using spatially-explicit individual tree growth models, a number of studies have demonstrated how species-specific competition changes with tree dominance or ontogeny Coates et al., 2009;D'Amato and Puettmann, 2004;Forrester et al., 2011;Gomez-Aparicio et al., 2011;Manso et al., 2015). In uneven-aged forests, dominance is highly, but not perfectly correlated with ontogeny. Rich vertical structure in uneven-aged stands is also a consequence of low stand density, and as stand density increases, vertical structure variation is lower. The documented effects make it necessary to explicitly consider effects of stand structure, or tree dominance for tree level studies, when studying the effects of stand density on productivity in mixed stands. We are not aware of any studies that explicitly addressed the interaction between stand structure, or tree dominance, and density on productivity in mixed species stands. We assume that effects of interspecific competition on stem diameter growth of individual trees in mixed species stands vary not only with neighborhood density but also with dominance of the focal tree, but are unable to formulate specific hypotheses for the three species interacting in the mixed stands we study.

Material
Long-term growth and yield monitoring plots in Switzerland  supplied the data for this study. For a large number of plots, tree positions of all trees with diameter at breast height (dbh) > 8 cm have been recorded and trees were repeatedly measured for dbh, allowing derivation of periodical increments. On some sites, a series of plots has been established. For our analyses, tree records were selected according to the following criteria. Only trees on the core plot were included, i.e. at least 10 m distance from plot borders, to allow quantification of spatially-explicit competition indices with full information about closest neighbors. Only trees with > 90% of the competition index (CIs) contributed by spruce, fir, or beech were included. Only sites with more than ten growth observations per species were used, because a lower number of observations might insufficiently cover the range of independent variables to allow for a reliable estimate of random site effects. Site 6003 was excluded from the analyses for beech because it had many observations and a large influence on the growth model for beech. Large negative basal area increments (< −10 cm 2 yr −1 ) were excluded.

Competition index
A spatially-explicit competition index was calculated, which considers effects of subject tree size, competitor size, distance, and species. For each competitor with a distance d (m) of max. 10 m and a size dbh c (cm), its contribution to the total competition for a given target tree is described as (1) and summed over all competitors per target tree as CIsum. The competition index is scaled to the size of the target tree (dbh t ) as which also scales CIs between 0 and 1. Similar competition indices have been used in earlier studies (Canham et al., 2004Pommerening et al., 2011;Sharma and Brunner, 2017), which estimated the parameters of the dbhand d-effect in Eq. (1) from the data. We set both parameters to values that have been reported in these studies to represent the effect for a range of different species. The choice of a fixed 10-m search radius for competitors was partly motivated by the results of those studies reporting strongest effects at shorter distances, and partly by the use of the distance function that predicts very small effects beyond a 10-m distance. The species proportion in the competition index is calculated as the proportion of ci contributions from that species to the total (CIsum) as The approach of representing effects of intra-and interspecific competition in the growth models is different from but inspired by similar approaches of earlier studies (e.g., Canham et al., 2004;Coates et al., 2013). Explicitly formulating species proportion as an independent variable allows testing for interactions of species proportion effects with competition indices (neighborhood density) and tree dominance (stand structure).

Description of data
The location of all experimental sites is shown in Fig. 1. In total, 83  sites were selected for model calibration, 26 sites were used in all three growth models because they contained a mixture with high proportions of all three species (Table 1). A total of 131 plots with 324 growth periods were selected. Most plots had sizes between 0.25 and 2 ha, with a total of 84.8 ha. Single-tree selection management was applied on 26 sites. Species composition varied from pure stands, to two-and threespecies mixtures with varying species proportions. Stand density varied between 20 and 60 m 2 ha −1 for most sites. Few stands were close to maximum density, the basal area relative to maximum basal area derived from a self-thinning line for spruce (Pretzsch and Biber, 2005), had a median of 0.49, 5 th percentile of 0.23, and 95 th percentile of 0.91. The distribution of the data over all independent and dependent variables (Table 2) showed a sufficient coverage, especially in the bivariate distribution of basal area and CIs to fit a response surface. For all species, a strong correlation between basal area and basal area growth existed on individual sites. In the pooled data from all sites, a large range of CIs for small trees allowed fitting the potential-modifier growth models. For the largest trees, competition varied little and CIs explained much less of the variation in growth. For the objective of our study, to test for density effects on species interactions, observations at high CIs, which represent high local density, are crucial to avoid extrapolating outside of the data when predicting growth in very dense stands. Observations at high CIs were well represented across all species proportions for most tree sizes, except for the largest trees. Growth records were from the period 1935 to 2018 for all species, but only after 1960 for silver fir. Most growth records were from the period 1980-2010. The length of the period covered by growth records varied from 4 to 62 years between sites.

Growth model
Basal area growth was used in this analysis, because diameter increment has been measured for all trees. Volume increment was avoided because it involves the uncertainties of estimated heights for most trees and the use of volume functions. Basal area increment is highly, but not perfectly, correlated to volume increment. The individual tree growth models are describing basal area growth (ig, cm 2 yr −1 ) as a function of initial basal area (g, cm 2 ). For this, a model form was chosen that allows for declining growth above a certain tree size (Eq. (4)). Growth is modified in a non-linear way by the competition index and the effect of the competition index was allowed to vary with tree size. Mixture effects on basal area growth were described as positive or negative linear effects as a function of species proportions. For the three-species mixture with little admixture of other species, two species proportions are sufficient to describe all species effects. Species proportions of spruce (S%) and beech (B%) were selected as variables (Eq. (3)). The model also allowed for interactions of the two species proportion effects. To test for density dependence of the mixture effects, interactions between species proportions and the competition index were added. Residual plots revealed that species effects varied with dominance of the trees expressed as relative height. rh is the height of the target tree relative to the mean height of all trees within a 10-m radius. As rh is strongly correlated with CIs and would in some cases compete with CIs in describing the competitive situation of the target trees, it is only used in the model combination with CIs to describe how species effects vary with dominance position, but not as a main effect.
where a to f are parameters to be estimated. The variable a i is a separate multiplier for each site i. It describes random effects that sites might have on the general level of growth, but also accounts for correlated error structures. During model fitting, a dummy-coded variable for each of the sites is used to estimate the set of a i . Time series for individual trees contain on average only two growth periods per tree (Table 3), which is the reason why no autocorrelation was included in the model.   Forrester Forest Ecology and Management 473 (2020) 118334 Variation in species interactions with site quality and climate were not considered in our analysis, despite some reports about that in individual tree growth studies of mixed stands (e.g., Condes and del Rio, 2015;Forrester et al., 2013). Model residuals did not indicate any strong interactions between species proportions and climate variables in our data.
Three models were fitted for the three species of the target trees. Model parameters were estimated using proc model in SAS with an ordinary least square method. Variables with insignificant (p > 0.05) parameter estimates were removed from the model one by one, starting with the variable with the highest p-value. Residual plots over independent model variables, other variables, and groups of variables were used to detect biased predictions of variable effects in earlier model versions.

Simulation of stand-level mixture effects at varying density
In order to demonstrate how species-specific competition effects on growth vary with stand density, we simulated growth of a single plot, using the given stand structure, but modified species identity to generate an even mixture of the three species. The three species-specific tree basal area growth models were applied to a series of generated plots at varying stand density. To derive species-specific competition effects, i.e. to compare growth in mixture with growth in stands with only conspecific neighbors, growth of each tree was predicted once with the species identity of neighbors in the generated mixture and once with species of the neighbors being the same as the focus tree.
Given that most of the data used to calibrate the growth models for the three-species mixture were from single-tree selection plots or other stands with large size variation, we could only simulate stands with varying density given a similar stand structure. Tree sizes and positions were therefore taken from a 1-ha subsection of a single-tree selection plot (site 1046), measured in 1980 (Fig. 2). Only tree sizes dbh and height were used from the original plot data, which follow the typical falling dbh distribution of single-tree selection stands (Fig. 3).
In order to simulate an even mixture of the three species, species identity from the original plot could not be used, because beech was only present in the understory of this particular plot. Species identity was therefore assigned using a row-wise mixture of the same trees in hexagonal spacing. A row-wise mixture in hexagonal spacing for three tree species means that two of the six closest neighbors are of the same species, i.e. it represents the most intimate species mixture. A hexagonal grid was established with a spacing to accommodate all trees of the observed plot. Trees were assigned to the closest grid cell center. In case of multiple trees assigned to the same grid cell, only the largest (by dbh) was assigned. The remaining trees were pooled in a list and sorted by descending dbh. Starting from the largest tree, these trees were assigned to the closest empty grid cell. The total basal area of this simulated mixture was composed of 37% spruce, 34% fir, and 28% beech (Fig. 4).
Stand density was varied during simulations by scaling the plot size, and ×, y of individual trees, to achieve a scaling of plot basal area (from the original 33.2 m 2 ha −1 , 425 trees ha −1 ). To allow calculation of CIs, only trees with a distance of at least 10 m from the plot border were used for growth predictions. The 10-m border was defined for the plot with the highest density, i.e. basal area factor of 3. The 338 core plot trees selected at this plot density where used to calculate results for all simulated stand densities, meaning that differences between stand densities are reflecting the effect on the same trees.
Given the single-tree selection structure of the simulated plot, increased density will change stand structure drastically in the long run. We therefore only varied stand density between 10 and 100 m 2 ha −1 . The extremes in that range might still represent an unrealistic stand structure.
For the site effect a i , the mean of all sites that contained all three species was used (8.56 for spruce, 1.51 for fir, 0.876 for beech), which is lower than the values estimated by the individual growth models for the given site, but likely more representative of ratios between species in a more even mixture.

Growth models
Individual tree basal area growth models for spruce, fir, and beech indicate significant variation of inter-and intraspecific competition with neighborhood density (CIs) and tree dominance (rh) ( Table 3). Growth model predictions for beech growing in pure beech neighborhoods show that potential growth only declines little even for the largest trees (dbh 100 cm), as can be expected for basal area growth models (Fig. 5). Site-specific variables a i were only weakly correlated between species (Table 1). This variable is, together with the other two shape variables of the basic growth model, determining the overall shape of the size-growth relationship. Therefore, mean values differ largely between the three species. In addition, the site-specific parameter a i might reflect site quality effects, but is also affected by the distribution of the data from that site over tree size and competition situation. In pure beech neighborhoods, the competition index reduces potential growth almost linearly (Fig. 5). However, for all species, the models describe highly non-linear effects of competition, which vary with species proportion and dominance.

Species-specific competition effects
The combined effects of all three species proportions on basal area growth for the ranges of CIs, rh, and g covered by the data are illustrated in Figs. 6-8. CIs and rh are strongly and non-linearly correlated in the data. Therefore, for three classes of CIs (0.1-0.2, 0.5-0.6, and 0.9-1.0), the 10 th , 50 th , and 90 th percentiles of rh and the median of g were calculated from the data and used for predictions. Growth levels, but not the species effects, changed when other percentiles of g were used for predictions with these models.
Variation in species proportions was unevenly covered by the data. For spruce and beech, many trees were from pure neighborhoods. Table 3 Parameter estimates and fit statistics of the tree basal area growth models (Eq. (4)). Standard errors of parameter estimates are given in parentheses for all parameters except for the site effect (a i ). Parameters are related to the variables tree basal area (g), competition index (CIs), spruce proportion (S%), beech proportion (B%), relative height (rh), and their interactions.  Forrester Forest Ecology and Management 473 (2020) 118334 Spruces were mostly mixed with fir, fewer with beech. Fir trees were mostly mixed with either spruce or beech. Beech trees were mostly in mixture with fir or spruce. However, for all three target species, all species neighborhood combinations were covered for trees over the full range of competition indices. Species proportion effects varied between target species, with CIs, and with rh. Effects were generally small for overstory trees with low CIs and high rh, but much stronger for understory trees with high CIs and low rh. The effects also varied much more with rh for trees with high CIs than for trees with low CIs.
Species-specific responses to admixture of other species vary largely with competition and dominance. For spruce (Fig. 6), intraspecific competition and competition from beech resulted in the highest growth in pure fir neighborhoods of trees with high and medium competition. Spruce trees with low competition, grew best in pure spruce neighborhood, irrespective of beech proportions. Fir was only a stronger competitor than beech in almost pure spruce neighborhoods. For fir (Fig. 7), trees with intermediate levels of competition grew best in pure spruce neighborhoods and worst in pure beech neighborhoods, irrespective of fir proportions. For fir trees under high competition, this  A. Brunner and D.I. Forrester Forest Ecology and Management 473 (2020) 118334 pattern was slightly modified into the highest growth at low beech proportions in combination with high fir proportions, especially for the least dominant trees. Fir trees with the lowest competition grew best in pure beech neighborhoods and worst in neighborhoods characterized by a combination of low beech and high fir proportions. For beech (Fig. 8), the strong intraspecific competition resulted in positive mixture effects. However, this mostly occurred in mixture with fir, which favors growth of beech trees with medium to high competition, and hampers growth of beech trees with low competition. For beech trees under strong competition, fir competitors had a slightly more positive effect than spruce competitors. Due to a considerable correlation between CIs and g for all three species, results described here for the three different levels of competition also indicate large differences in tree size, as indicated in the figures.

Simulation of stand-level mixture effects at varying density
Simulation results for the total stand ( Fig. 9) indicated positive effects of mixture compared to pure stands, with increasing overyielding as stand density increased, reaching 18% at the highest simulated density. However, the species-specific responses deviated significantly from this general pattern. Whereas beech shows a very strong overyielding that increases with stand density, fir shows a constant low overyielding. For spruce, underyielding at low density shifts into overyielding at higher density.
The pattern of monotonically increasing growth with stand density contrasts with the common degressive shape of growth-density relationships as maximum stand density is approached. Given the large size variation in this single-tree selection plot, the variation of density has very different effects on overstory trees compared with understory trees. While the increased density at the two extreme densities strongly increases CIs and reduces growth for trees of dbh < 30 cm, it has very little effect on CIs and growth of the largest trees (Fig. 10).
Despite their lower numbers (23% of all trees, Fig. 3), trees > 30 cm dbh contribute 83% to the basal area and 70% to the basal area growth at the original density. It is therefore interesting to see how these two classes of trees reacted to the increased density (Fig. 11). Given the small increase in CIs of large trees, it is not surprising to see that mixture effects are much lower for large trees than for small trees. Trees with dbh < 30 cm also show a much more asymptotic pattern of growth over density than reported in Fig. 9 for all trees (not shown).

Effects of stand density
Our simulation shows strong effects of stand density on productivity in mixed stands. Overyielding was strongest at the highest stand density and reduced to underyielding at the lowest stand density (Fig. 9). This result is only valid for the given stand structure of a single-tree selection stand, and the very intimate spruce-fir-beech mixture. However, neighborhood density effects in the individual tree growth models indicate a more general pattern of much stronger inter-and intraspecific interaction at higher density for all three species (Figs. 6-8) across the range of studied stand structures. Our simulations also illustrated how the contribution of the three species to the total overyielding varied with density from underyielding to overyielding. Also, trees of different sizes, and dominance, contributed very differently to the total overyielding. These simulations with individual tree growth models therefore help to explain stand-level patterns, and can also be used to study Fig. 6. Effects of species composition on basal area increment (isolines, cm 2 yr −1 ) of spruce trees of three different classes of competition index (CIs) and basal area (g, cm 2 ) displayed in columns, and varying relative height (rh) displayed in rows within columns (site effect a i = 8.25). Fir proportion varies from 1.0 at the origin to 0.0 along the line connecting 1.0 of the other two species proportions. All species proportions are for tree neighborhoods, quantified by contributions to CIs.
how different species and tree size compositions will change stand-level mixture effects. Simulations in this study only predict short-term growth responses reliably. Long-term stand development will also be affected by changes in stand structure and mortality, especially at higher density. Over longer periods, the reported density effects might therefore change considerably. Regardless of specific density responses of mixture effects depending on period length, stand structure, species identity, or site conditions, stand density is likely to have a significant effect and needs to be controlled in experiments or sampling, or included as a factor in the analyses (Kelty and Cameron, 1995;Vanclay, 2006a). Our results also indicate that mixture effects can be minimal in very open stands and therefore not relevant for stands managed with heavy thinnings over longer time periods.
As discussed in the introduction, few studies have explicitly addressed the effects of stand density on mixture effects. No results are available for a realistic stand structure in a mixed stand of three shadetolerant species in older stands. Our hypothesis was therefore based on individual effects observed in other studies. Data, growth models, and Fig. 7. Effects of species composition on basal area increment (isolines, cm 2 yr −1 ) of fir trees of three different classes of competition index (CIs) and basal area (g, cm 2 ) displayed in columns, and varying relative height (rh) displayed in rows within columns (site effect a i = 1.55). Fir proportion varies from 1.0 at the origin to 0.0 along the line connecting 1.0 of the other two species proportions. All species proportions are for tree neighborhoods, quantified by contributions to CIs.
simulation results support the hypothesis of a strong effect of stand density on mixture effects.
Intra-and interspecific interactions on basal area growth have been described for each of the three species (Figs. 6-8). These interactions varied in some cases largely with neighborhood density and dominance in this mixture, indicating significant effects of stand structure on species interactions. Comparison with other studies would therefore only be possible if stand structure is similar or effects were quantified in a similar way. However, a few more general patterns are comparable to earlier results such as Perot and Picard (2012), where intraspecific competition was considerable for all three species. Similar to our results, beech increased growth with increasing admixture of a less shadetolerant conifer species, but only at higher stand densities (Condes et al., 2013).
To analyze the variation of mixture effects with stand density, we used a method that was frequently applied by other studies, i.e., individual tree growth models that quantify the contribution of all competing species using spatially-explicit competition indices. While earlier Fig. 8. Effects of species composition on basal area increment (isolines, cm 2 yr −1 ) of beech trees of three different classes of competition index (CIs) and basal area (g, cm 2 ) displayed in columns, and varying relative height (rh) displayed in rows within columns (site effect a i = 0.78). Fir proportion varies from 1.0 at the origin to 0.0 along the line connecting 1.0 of the other two species proportions. All species proportions are for tree neighborhoods, quantified by contributions to CIs. studies used these growth models to test hypotheses about intra-and interspecific competition at the tree level (e.g. Coates et al., 2009;del Rio et al., 2014), we used them to predict stand-level mixture effects under varying stand density. These simulations allowed us to compare the same trees in the same stand structure growing in mixtures and with intraspecific competition only. This comparison is impossible to achieve in experiments. Variation in stand density of a realistic stand structure further allowed us to test for the effects of density on mixture effects in a setting that cannot be generated by experiments. These simulations illustrate the contrast between intra-and interspecific competition for a range of tree sizes and dominance positions, which have been shown in growth models to vary in their interspecific competition. They also illustrate the contribution of groups of trees of different sizes and dominance to the total stand response. We have not found any other study that has addressed the question of stand density effects on productivity in mixed stands with a similar method. Two studies reported on similar methods that applied individual tree growth models in simulations to study stand-level patterns in mixed stands. Vanclay (2006b) illustrated a similar method to address density effects with a preliminary growth model. Perot and Picard (2012) used analytical solutions of their individual tree growth models to find optimal species proportions for a set of experimental plots.
Studies addressing specific processes and structures affecting productivity in mixed stands have mostly used two-species mixtures, while mixtures of more than two species have mostly been used to study the effects of species richness (Forrester and Bauhus, 2016). Studying detailed effects of stand density and stand structure on species interactions in a three-species mixture requires new methods to be employed. Individual tree growth models have earlier been used to study effects of species mixture on stand-level growth in mixtures of more than two species (e.g. Mina et al., 2018a). We applied these methods to data from a large series of long-term observation plots where individual trees were selected that experienced most of the competition from any given combination of the three species. We are not aware of other studies applying a similar method.
Compared to many earlier studies of mixture effects, the data in this study are from many different sites, represent large plot sizes, and cover a large range of stand ages and tree sizes. Stand structure also varies considerably between the plots, with many plots being managed by the single-tree selection system. Tree growth models fitted to these data therefore cover a wider range of individual tree conditions in terms of density, mixture, and dominance, compared to earlier studies.
In this study, we describe how mixture effects on tree and stand basal area increment vary with stand density and stand structure. A large number of processes and patterns are causing these mixture effects, which vary in space and time (Forrester and Bauhus, 2016). Basal area increment is an integrating measure that might in some cases reflect the sum of opposing effects of inter-and intraspecific competition. Given that this study is only based on the tree measurements, we refrain from speculating about individual processes and patterns causing the observed mixture effects.

Effects of stand structure
Given the few studies that analyzed effects of stand structure on mixture effects, we were unable to formulate specific hypotheses for the three species in our study. The strong variation of species effects with dominance in our results indicates that this effect needs to be considered in experiments and analyses of mixture effects. Simulation results also illustrate large variation in mixture effects between tree size groups (Fig. 11), which describe stand structure effects for the given single-tree selection plot. Changes in the relative dominance of a particular species in mixed stands may result in changes in the importance of neighborhood interactions over time (D'Amato and Puettmann, 2004). Manso et al. (2015) also demonstrated how interspecific competition and overyielding depends on stand structure and individual tree dominance (their Fig. 3). Similar to our results, diameter growth of suppressed beech trees was much more affected by interspecific competition than growth of dominant trees. Perot and Picard (2012) also demonstrated that stand structures with groups of oaks in mixture with pine result in less overyielding due to strong intraspecific competition of oak in these groups.
Growth models fitted to national forest inventory data from Switzerland for the same species mixture as in our study (Mina et al., 2018a, their Fig. 2) showed similar patterns of interspecific competition among the three species (Figs. 6-8 for dominant trees with medium CIs). Also, in their study, interaction between the two other admixed species explained part of the variation in basal area growth. Interactions between species composition and competition being modified by relative height in a very significant way in our models for all three Fig. 9. Effect of stand density on basal area growth in simulated mixed stands. The basal area increment ratio in the lower panel is the basal area increment in the simulated mixed stand relative to the theoretical growth without mixture effects but identical species composition. The ratio per species is for the sum of all trees of that species (not the mean of individual trees). species might be similar to the results of Mina et al. (2018a), who demonstrated how species effects for beech can differ with basal area of larger trees (BAL). Even though they interpret the effect of the two spatially non-explicit competition indices as mainly indicating competition symmetry, these indices describe at the same time the position of focal trees in the vertical stand structure, and therefore indicate stand structure effects as well. Another set of growth models fitted to a wider sample of the national forest inventory data from Switzerland reported how the ratio of intra-and interspecific competition (complementarity) changed with the strength of the competition (BAL and stand density index, SDI) (Mina et al., 2018b). For spruce at increasing BAL, individual tree basal area growth was increasingly reduced in mixed stands compared to pure stands in all mixture types with fir and beech. SDI also affected complementarity but only in the spruce-beech mixture, reducing mixture effects as SDI increased. For fir, this effect was only present in mixture with beech. For beech, no effect of BAL on complementarity was detected. Given that BAL only describes part of the competition, it is difficult to compare those results with the detailed effects of interspecific competition reported in our results.

Species interactions in spruce-fir-beech stands
Simulation results with the given plot effects, stand structure, and intimate species mixture indicated overyielding at the stand-level in spruce-fir-beech stands at higher density. The overyielding was mostly caused by increased growth of beech in mixture. This result confirms other studies of mixture effects in spruce-fir-beech stands. Pretzsch and Forrester (2017, p. 148) reported overyielding at the stand level on average, but large variation between individual experimental plots. In another study on the stand level, variation in growth could be partly explained by variation in stand structure, without completely depicting the specific interactions between stand structure and species composition (Torresan et al., 2020). Analyzing the detailed interactions between the three species in two-and three-species mixtures in Switzerland, the strong competition from beech on the two other species was demonstrated (Mina et al., 2018a). Growth in spruce-fir-beech stands from a network of permanent sample plots (Hilmers et al., 2019) indicated strong negative growth trends of spruce between 1970 and 2010, and positive trends for fir during the same period. These trends are most likely explained by the severe decline of fir between 1970 and 1990, when spruce experienced less competition from fir. Our data also covered mostly the period between 1980 and 2010 in a region that has been affected by fir decline. It is therefore possible that positive effects of fir neighbors on spruce and beech in our models only reflect the species interactions in that period and cannot be extrapolated to other periods.
A number of studies have analyzed the growth of two-species mixtures containing spruce, fir, or beech, all of which differ little in shade tolerance. For the spruce-fir mixture, interspecific competition between the two species has been reported to be rather close to intraspecific competition for both species, but this ratio varied a lot with site conditions, causing some overyielding under most conditions (Forrester et al., 2013). Fir contributed mostly to the observed overyielding on the stand level, which was also observed in studies in France (Toigo et al., 2015;Vallet and Perot, 2011). Also in Switzerland, stand-level growth in this mixture varied largely with site conditions, often resulting in underyielding in mixed stands (Huber et al., 2014). Similar to our results (Figs. 6 and 7), spruce and fir facilitated each other based on data from the Swiss national forest inventory (Mina et al., 2018a(Mina et al., , 2018b. The spruce-beech mixture has frequently been studied and overyielding has only been reported on some sites (Pretzsch et al., 2010;Pretzsch and Schütze, 2009). Swiss national forest inventory data indicated mostly overyielding in this mixture, mainly caused by beech, but also varying largely with site conditions (Houpert et al., 2018). Overyielding in this mixture has been reported to be mostly due to beech (Pretzsch et al., 2010;Toigo et al., 2015). The strong intraspecific competition of small to medium sized beech trees in our data (Fig. 8) confirms these results. Few studies have reported on mixture effects in the fir-beech mixture, but also here Toigo et al. (2015) reported the overyielding to be mostly due to beech. del Rio et al. (2014) reported that competition between beech and fir was not very species-specific, and mostly depending on relative size.

Conclusion
This study demonstrated that tree species mixture affects basal area growth only at higher stand density in this stand type, defined by the spruce-fir-beech mixture and the given stand structure. The strong interaction of mixture effects with stand density are likely more general for other mixtures and stand types as well and therefore needs to be considered in experimental designs, sampling designs, and management plans. The study also demonstrated that this stand-level question can be addressed with individual tree growth models. These or other growth models can be used in different scenarios in future studies to analyze under which specific conditions, e.g. species proportions and stand structure, interspecific competition affects stand-level growth.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Fig. 11. Effect of stand density on basal area increment ratio in simulated mixed stands for trees below or above dbh = 30 cm. See Fig. 9 for definition of the basal area increment ratio.