Modeling Diameter Growth of European Beech in Mixtures with Various Tree Species: The Impact of Size-Symmetric and Size-Asymmetric Competition

Abstract

Mixed forests provide several ecosystem service benefits, and they also often show higher productivity than pure forests. In mixed forests, several interactions among tree species occur, with size-symmetric and size-asymmetric competition being particularly important. We studied diameter growth of European beech in pure stands and in mixtures with oak, maple, pine, spruce, fir, and spruce and fir combined on extremely diverse beech sites in Slovenia, using forest inventory sample plots (n = 26,793, 500 m² each). For each mixture, we developed models of 10-year individual tree diameter increment (id) using natural splines and incorporating tree, competition, stand, site, and climate variables that were mainly gathered in regular forest inventories. Competition was represented using simple indices: stand basal area (BA) for size-symmetric competition, basal area of overtopping trees (BAL) for size-asymmetric competition, and reduced competition due to harvesting (CUT). The models revealed differences among mixtures and a strong influence of competition. Id was among the lowest in pure stands and substantially higher in mixtures, indicating strong intraspecific competition. Overall, size-symmetric competition was more influential, but size-asymmetric competition appeared important in some mixtures. We recommend growing beech in mixtures with other species and applying a forest management approach that accounts for competition symmetry, which proved crucial in each mixture.

1. Introduction

Approximately 23% of the land in the pan-European region is covered by mixed forests [1], which have been recognized as having several advantages over pure forests and monocultures. They can provide a broader range and higher levels of ecosystem services [2,3,4]. Forests with higher structural and species diversity are often more resistant and resilient to biotic and abiotic disturbances [5,6,7]. Finally, mixed forests can also exhibit higher productivity in certain environmental conditions [8,9,10].

In mixed forests, several interactions among tree species occur simultaneously. Interactions between species growing in mixtures are often described in terms of competition, competitive reduction, and facilitation [11]. Competition occurs when interactions between species or populations reduce another species’ functioning, survival, or growth. Competitive reduction occurs when interspecific competition impacts growth less than intraspecific competition [9,12]. Finally, facilitation occurs when at least one species positively influences the functioning and growth of another [9], which can happen due to, for example, increased nitrogen fixation [13] or hydraulic lift [14]. In natural mixed forests, processes can be described in terms of species complementarity, reflected in changes in the growth of individual trees or entire stands [9,10,11,15]. Trees in mixtures tend to be more productive if they use site resources more efficiently due to niche differentiation in ecological processes, such as nitrogen fixation, increased light absorption, and increased water-use efficiency. This may reflect differences in height growth patterns, phenology, crown and root architecture, and resource-use strategies [4,8,16,17]. Competition works differently in mixed forests than in pure ones since tree species occupy growing space differently. In addition, the allocation of resources such as light, water, and nutrients among species is different. When there is a strong niche complementarity between tree species, interspecific competition may be lower than intraspecific competition [18,19].

To address the effect of mixture on the diameter growth of individual trees, mixing effects should be analyzed through possible growth deviations in mixtures compared with pure forests [5,10,20]. Trees are classified into mixture classes based on neighboring trees, e.g., [21], or based on the BA proportion of admixed species, e.g., [10,22].

According to the limitation-caused matter partitioning (LCMP) hypothesis, competition and its prevailing mode are influenced by the dominant limiting resource, for example, light, water, or nutrients [23,24]. Size-asymmetric competition mode arises when larger individuals claim resources disproportionately to their size, leaving fewer resources available to smaller individuals; a classic example is competition for light. In contrast, the size-symmetric competition mode occurs when competitors divide resources evenly [25]: for example, competition for water and belowground soil resources [24,26].

Mixture and competition should be studied together, and species proportion can be used as a proxy for the partitioning of growing space [15]. Different metrics and procedures have been used in studies to express and describe different complementary interactions on individual tree or stand growth. Sometimes the competition status of a tree is included as a modifier component that reduces the potential growth rate of a tree [15,17,18]. Various competition indices based on tree-to-tree distances, crown dimensions, stand density, or detailed spatial information have often been used to characterize size-symmetric and size-asymmetric competition [18,24,26,27,28,29]. However, for practical forestry, simple measures of competition are especially welcome, as competition indices are computationally demanding. Simpler stand-level metrics, such as stand basal area (BA), a proxy for belowground resource availability and thus size-symmetric competition, and overtopping basal area (BAL), a surrogate for size-asymmetric light-related competition, have already been used in ecological modeling [22,30,31].

European beech (Fagus sylvatica L.; hereafter beech) is one of the most important tree species in Europe, currently covering approximately 12 × 10⁶ ha [32] and providing high ecological and economic value in Central European forestry [33]. Due to its strong competitiveness, broad ecological amplitude, and capacity to recover after disturbances [7], it dominates late-successional forests [34]. However, beech commonly form mixed stands with various tree species, namely Norway spruce (Picea abies (L.) H. Karst., hereafter spruce), silver fir (Abies alba Mill., hereafter fir), pines (Pinus spp., hereafter pine), oaks (Quercus spp., hereafter oak), maples (Acer spp., hereafter maple), and many other species. Multiple studies have reported enhanced beech growth in mixtures with different species. Beech may exhibit a facilitative relationship and benefit from higher growth rates wen mixed with species such as Scots pine (Pinus sylvestris L.) [35], spruce [36], Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco) [37], fir [28], or conifer species [31].

For modeling the growth of individual trees, several approaches have been identified. To predict resource and growth partitioning among trees in a stand, site-specific potential growth–size relationships for individual trees can be developed, reflecting growth rate under optimal growth conditions [24]. This potential growth is subsequently reduced to the expected growth rate by means of various modifiers, reflecting the competition status of the observed tree [22]. The second approach is to develop individual tree growth models by testing the relevance and contribution of many potential explanatory variables, e.g., [18,28,31,38,39,40]. The objective is to determine the main predictors of tree growth; thus, it is crucial to test the relevance of all meaningful variables related to tree characteristics, competition, and stand, site, and climate conditions.

Tree and stand characteristics have often been identified as the main predictors of individual tree growth [29,39], but competition indices, e.g., [24,26,41], or proxies for competition, e.g., [18,22,31], have also been shown to be crucial for the growth of individual trees. The influence of size-symmetric and size-asymmetric competition on tree growth was found to be divergent and differed by tree species composition and forest type, e.g., [10,15,18,31]. Among tree variables, tree diameter or its transformations [29,42,43] have frequently been recognized as a crucial predictor.

Explanatory stand factors usually encompass parameters related to stand density, structural diversity, and tree species mixture [44]. Stand density can be described by the stand density index (SDI), number of trees, or stand basal area. Stand density most frequently indicates a negative impact on tree growth. Structural diversity, indicated by various indices, such as the Gini index [45], influences tree growth through competition, which differs between diverse stand structures, e.g., [36]. With respect to species mixture, beech has often been reported to grow better in mixed than in pure stands [39,46]. This has been confirmed in mixtures with pine [27,35], oak [38], or spruce and fir [31,46]; however, del Río et al. [18] found that there were no complementary effects between beech and fir in forests in Spain.

Beech diameter growth is also affected by climate variables and climate change in complex ways. Studies report contrasting growth trends, with some findings showing increasing growth [36,47] and some showing decreasing growth [48,49]. Growth reductions have been observed, particularly in Central Europe [41] and Mediterranean regions [50]. Repeated drought events clearly reduce beech growth [38,51], and simulations forecast further decline, especially at the southern edge of its distribution [49]. However, mixing beech with other species may reduce drought-related declines in tree growth [9].

Beech is currently an important species and is expected to remain so in the future. Understanding its functioning and growth in pure and mixed forests is crucial for adapting forest management to changing climatic conditions [52]. Identifying conditions under which mixed forests enhance beech functioning and growth is particularly crucial. Slovenia offers favorable conditions for such a study. Beech is the most common tree species in Slovenia, accounting for 33% of the national timber volume and covering approximately 70% of the forested area [53]. Most of these forests are mixed and mainly dominated by beech, providing an appropriate context for investigating the diameter growth of beech trees in pure forests and in mixtures with other species. Despite its relatively small area, Slovenia exhibits a highly heterogeneous landscape, encompassing the Alps, the Mediterranean region, the Dinaric Mountains, and the Pannonian Basin, covering a large altitudinal gradient (0 to ~1750 m). Furthermore, the broad gradients of climatic, edaphic, and stand conditions provide an additional strong foundation for investigating the growth of beech trees in pure and mixed forests. Our research objectives were (i) to develop individual tree diameter increment models for beech in pure stands and in various two-species and three-species mixtures growing on beech forest sites, using only explanatory variables for which data are collected during regular forest inventories or can be derived from publicly available databases, and to assess whether the explanatory variables included in the models, as well as the magnitude and direction of their effects, differ among mixtures, and (ii) to inspect the influence of BA and BAL as indicators of size-symmetric and size-asymmetric competition across different mixtures to support the adaptation of forest management to the symmetry of competition within each mixture.

2. Materials and Methods

2.1. Study Area, Species Mixtures, and Data

Slovenian forests (in total 1.2 million ha; forest cover of 59%) are characterized by substantial variation in tree species mixture, stand structure, and geoclimatic conditions. Climatic regimes range from Sub-Mediterranean in the southwest to temperate humid in the central region and the continental climate in the northeast (

Table 1. Definition of beech mixtures analyzed and tree species’ proportions, sample sizes (N trees and N plots), and mean values of basic tree, stand, site, and climate characteristics for all mixtures (dbh—mean diameter of beech trees used for modeling, BA—stand basal area, D_DOM—mean diameter of the 100 thickest trees on a plot (i.e., dominant stand diameter), QMD—quadratic mean stand diameter, SProd—site productivity (for details see explanation in Table 2), ELE—elevation, and Tspr—mean spring temperature in the period March–May).

Mixture	Beech Share	Admixed Species	N Trees	N Plots	dbh (cm)	BA (m2 ha−1)	DDOM (cm)	QMD (cm)	SProd	ELE (m a.s.l.)	Tspr (°C)
OAK1	60%–90%	oak (Quercus petraea, Q. robur)	9275	835	27.1	32	38.4	25.7	2.011	437	8.5
OAK2	30%–60%		2736	374	24.8	33.3	38.7	25.5	1.965	436	8.5
OAK3	<30%		719	246	22.7	32.5	38.5	26.5	1.968	463	8.6
MAP1	60%–90%	maple (Acer pseudoplatanus, A. platanoides)	9750	967	28.3	30.9	38.0	26.4	1.994	809	7.1
MAP2	30%–60%		849	149	27.5	27.8	36.2	25.7	2.044	783	7.2
MAP3	<30%		70	26	20.9	26.7	34.2	23.6	2.003	711	8.0
PIN1	60%–90%	pine (Pinus sylvestris, P. nigra)	1231	119	24.8	29	36.7	24.3	1.908	392	8.9
PIN2	30%–60%		4954	701	23.9	31.5	37.5	24.9	1.797	369	9.7
PIN3	<30%		3672	1338	19.5	33.3	35.0	23.7	1.791	383	9.3
SPR1	60%–90%	spruce (Picea abies)	12,924	1187	26.6	32.3	38.3	25.6	1.936	761	6.9
SPR2	30%–60%		6888	988	26.3	34.2	40.7	27.2	1.944	770	7.3
SPR3	<30%		4739	1768	23.0	37.5	41.9	28.6	1.942	807	7.5
FIR1	60%–90%	fir (Abies alba)	6228	665	30.4	32.1	42.3	29.1	1.984	934	6.8
FIR2	30%–60%		2900	504	30.4	32.4	45.1	31.0	1.967	881	6.9
FIR3	<30%		1294	484	25.9	34.5	47.4	32.0	1.996	826	6.9
SPFI1	60%–90%	spruce & fir (Picea abies, Abies alba)	3957	347	28	35.5	40.7	26.7	1.972	979	6.7
SPFI2	30%–60%		690	98	27.4	35.8	43.0	28.2	1.993	895	7.0
SPFI3	<30%		442	142	22.7	37.3	43.0	28.5	1.982	860	7.1
BEE	>90%	-	58,587	4650	28.4	31.8	38.1	26.8	1.978	727	7.3

). The relief is highly diverse, featuring plains and low hills to predominantly rugged hilly terrain with slopes of varying steepness to high mountains. The altitude of examined forests ranges 0-~1750 m, and the bedrock is highly diverse, ranging from predominant carbonate rocks (mainly limestone and dolomite) to various silicate rocks, including magmatic rocks, schists, and sandstones. The dominant soil types include leptosols, eutric and chromic cambisols, and dystric cambisols [54].

The average stand volume of forests in 2014, the last year of our dataset, was 304 m³/ha, and the mean annual volume increment was 7.4 m³/ha [55]. Forests are managed, and close-to-nature forestry practices, relying predominantly on natural regeneration, have been implemented for decades and, in some areas, for more than a century. Suppressed understory trees and advanced regeneration of various tree species, as well as shrubs, are integral parts of these forests. The most widely applied silvicultural systems include irregular shelterwood, group selection, and single-tree selection [56].

Based on potential natural vegetation, beech forests cover approximately 70% of the forested area [53]. Beech is the most abundant species in the stand volume (33.0%), followed by spruce (30.0%) and fir (7.5%) [55]. Moreover, our dataset encompasses a wide spectrum of developmental stages, stand densities, and combinations of site characteristics (Table 1), providing a comprehensive basis for modeling diameter growth across diverse conditions.

More than 98,000 circular fixed-area permanent sampling plots (PSPs) were established by the Slovenia Forest Service in the 1990s [55] and have since been remeasured every 10 years, with roughly 10% of PSPs surveyed each year. The PSPs mainly form a systematic sampling grid of 0.25 × 0.50 km, but, in areas of highly intense forest management, also of 0.20 × 0.50 km or 0.25 × 0.25 km. On PSPs, trees with a diameter at breast height (dbh) between 10 and 29.9 cm are measured within a 200 m² subplot (radius = 7.98 m), while those with dbh ≥ 30 cm are recorded within a larger 500 m² subplot (radius = 12.62 m) [57]. The first measurements in our database were conducted in the period of 1994–2004, and the first remeasurements were conducted in the period of 2003–2013 [55]. An example of the dataset is shown in Table S1.

The study focused on the diameter growth of beech in homogeneous stands within beech forest types. The initial database comprised 193,314 individual beech tree data (dbh ≥ 10 cm) collected from 26,793 PSPs (Figure 1). Only structurally homogeneous plots, defined as homogeneous in tree sizes (i.e., even-sized) according to the Gini index [45], were included (n = 15,558). The Gini index, which is frequently used in forestry to assess stand structural homogeneity or heterogeneity [45,58], was calculated at the plot level based on tree basal area [59]. To classify forests into homogeneous and heterogeneous categories, k-means cluster analysis was performed using a Gini index threshold of 0.33 [58], which is consistent with the relevant literature [45].

Following the study objectives, we modeled the diameter growth of beech in pure stands and mixtures with several other tree species. Altogether, 131,905 individual beech trees were analyzed and used for modeling, while an additional 99,846 trees of admixed tree species were included in calculations of all competition and stand explanatory variables. Mixtures were defined based on (i) the proportion of beech in the basal area in each plot and (ii) the dominant admixed species. Accordingly, we considered pure beech plots (≥90% beech) and mixtures where the proportion of beech was <30%, 30%–59.99%, and 60%–89.99%, and the proportion of an admixed species (in one case, two admixed species) was at least 80% of the remaining BA. In addition to pure beech stands, we addressed six main mixtures in which beech is mixed with oak (Quercus robur L., Q. petraea (Matt.) Liebl.maple (Acer pseudoplatanus L., A. platanoides L.), pine (Pinus nigra JF Arnold, P. sylvestris L.), spruce, fir, and spruce and fir combined (spruce and fir). In total, we modeled beech growth in 18 mixtures and in pure beech stands (Table 1 and Table S2). For each mixture, we ensured a sufficient number of sample plots to avoid problems of overparameterization and convergence due to singularity errors [10].

2.2. Growth Modeling Procedure

2.2.1. Diameter Increment as the Response Variable

When measured in the field, tree dbh values were rounded down to the nearest centimeter [57]. Therefore, both the first and second dbh measurements were adjusted by adding a random component ρ, uniformly drawn from the interval [0, 1] for trees with a positive increment. The 10-year diameter increment (id) was then computed as the difference between these corrected dbh values [60], representing the dependent response variable. For trees with an observed zero increment, id was set to the random component ρ, while negative increments were truncated to zero under the assumption that they resulted from measurement error. Extremely large increments (id > 15 cm) were interpreted as errors and excluded from the modeling procedure [43]. After data cleaning, id ranged from 0 to 15 cm 10y⁻¹, with a mean of 2.9 cm 10y⁻¹. To analyze the diameter growth of beech across mixtures, analysis of covariance (ANCOVA) with dbh as a covariate was applied, and the estimated marginal means were plotted and compared. To reduce skewness and heteroscedasticity in the models, a square-root transformation of id was applied in the modeling procedure [61].

2.2.2. Explanatory Variables as Potential Predictors

In the modeling procedure, five groups of potential explanatory variables were included, namely tree, competition, stand, site, and climate explanatory variables (Table 2). Among tree variables, we used dbh and the quotient dbh/QMD, which indicates the position of the tree within the diameter distribution on a plot, e.g., [43].

To express competition among trees, three variables were used. The first was overtopping basal area (BAL), defined as the basal area of trees larger in dbh than the subject tree [40]. BAL is used to characterize size-asymmetric competition among trees on a plot, e.g., [30]. Stand basal area (BA) was applied to indicate size-symmetric competition among trees [22]. To detect the influence of release effects and reduced competition due to forest management, the relative basal area of harvested and naturally dead trees (CUT) was applied, indicating the impact of silvicultural measures and mortality. It was defined as the ratio between the BA of harvested and dead trees between two consecutive measurements and the BA of the stand at the first measurement.

Five stand variables were applied in the modeling procedure. Dominant stand diameter (D_DOM), calculated as the mean diameter of the 100 thickest trees per ha, and quadratic mean diameter (QMD) both indicate the development phase of a stand represented by each plot. Although only homogeneous stands were included in the analysis, we used the Gini coefficient (GINI), calculated at the plot level, to describe tree size diversity, as it proved to be a good indicator of stand structure [45,59]. In addition, we also applied the maximum diameter of trees on a plot (D_MAX), expressing forest management goals related to target tree size.

Among site variables, we included twelve, namely, elevation (ELE), slope (SLP), aspect (ASP), site productivity (SProd), variables describing soil characteristics (DSoil, DSoil_A, pH, pH_A, ORG, ORG_A, and SoilT), and beech forest types (BeechT), which are important tools for ecological forest management planning in Slovenia, allowing site-specific planning. As a proxy for SProd, a coefficient K was used, representing the volume of a tree with a reference dbh of 45 cm. K is applied in the one-parameter tree volume functions used in Slovenia to calculate tree volume [62]. It ranges from 1.20 to 2.95 m³, reflecting differences between sites in terms of the height of a tree of the same dbh.

Seventeen climate variables were tested for inclusion in the models. Temperature-related variables (T, Tmax, Tmin, Tmax_Jul, Tmin_Jan, Tspr, Tspr_sum, BIO2, BIO10, and BIO11), solar radiation (SOLRAD), and annual precipitation (PCP) were derived using long-term climate records [63] and the SLOCLIM high-resolution daily climate dataset for the period of 1950–2018 at 1 km² resolution [64]. To calculate the standardized precipitation evapotranspiration index (SPEI), daily historical climate data of precipitation and evapotranspiration over Slovenia on a regular grid with a resolution of 0.125° were obtained for the period of 1981–2010 [63]. The SPEI was then calculated using the SPEI package in R, v1.8.0 [65]. The SPEI indices were calculated using data available up to 2010. For plots where the second measurement was conducted after 2010 (i.e., in 2011–2014), we assigned the SPEI value corresponding to the latest available year, namely 2010.

Table 2. Description of explanatory variables included in the diameter increment modeling procedure; model: x means the variable was included in the model calculation, and mc means the variable was excluded from the modeling procedure to avoid multicollinearity.

Type	Acronym	Description	Unit	Mean	SD	Min	Max	Model
Dependent	id	Diameter increment at breast height	cm	2.9	2.1	0	14.5
Tree	dbh	Diameter at breast height	cm	26.8	11.8	10	88	x
	dbh/QMD	Quotient dbh/QMD						mc
Competition	BAL	Overtopping basal area	m2 ha−1	18.49	12.06	0	89.22	x
	BA	Stand basal area	m2 ha−1	32.02	10.53	0.87	90.54	x
	CUT	Relative basal area of cut and dead trees	%	10.1%	13.5%	0.0%	99.4%	x
Stand	N	Number of trees per hectare		698	381	40	3000	mc
	QMD	Quadratic mean diameter	cm	26.2	7.2	10.0	68.8	mc
	DDOM	Dominant diameter as mean diameter of the 100 thickest trees per ha	cm	38.6	8.7	10.0	75.3	x
	DMAX	Maximum plot diameter	cm	44.7	10.4	10.0	88	mc
	GINI	Gini index (heterogeneity)	%	0.26	0.05	0.00	0.33	x
Site	ASP	Aspect	1-warmer aspects S, SE, SW, W; 0-colder aspects N, NE, NW, E					x
	SLP	Slope	°	20.1	9.7	0	54	x
	ELE	Elevation	m a.s.l.	688	294.5	80	1629	x
	SProd	Coefficient K as a proxy for site productivity	m3	1.97	0.26	1.20	2.95	x
	DSoil	Depth of soil (cm)	cm	59.3	24.5	0	365	x
	DSoil_A	Depth of soil A horizon	cm	13.5	5.1	0	57	x
	pH	Soil pH (average)		5.12	1.11	0.00	7.60	x
	pH_A	Soil pH of the A horizon		5.07	1.33	0.00	7.50	x
	ORG	Sum of organic matter	%	8.3	3.3	0	38	x
	ORG_A	Sum of organic matter in A horizon	%	3.9	0.9	0	8	x
	BeechT	Beech forest type	Categorical variable *					x
	SoilT	Soil type	Categorical variable *					x
Climate	T	Average annual temperature	°C	8.0	1.5	0	11	mc
	Tmax	Maximum annual temperature	°C	12.8	2.1	7.0	18.5	mc
	Tmin	Minimum annual temperature	°C	3.7	1.5	−1.0	7.0	mc
	Tmax_Jul	Maximum temperature in July	°C	23.0	2.3	16.0	28.0	mc
	Tmin_Jan	Minimum temperature in January	°C	−4.2	1.5	−9.5	1.5	mc
	Tspr	Average temperature for March, April, and May	°C	7.5	2.0	1.0	12.0	x
	Tspr_sum	Sum temperature for March, April, and May	°C	22.4	5.9	3.0	36.0	mc
	SOLRAD	Solar radiation	kJ m−2	1907.5	105.6	1580	2395	mc
	BIO2	Mean diurnal range (Tmax–Tmin)	°C	9.1	2.4	0.0	16.0	mc
	BIO10	Mean temperature of warmest quarter (°C)	°C	16.4	2.0	10.3	21.8	mc
	BIO11	Mean temperature of coldest quarter (°C)	°C	−0.6	1.3	−4.3	5	mc
	SPEI6Sept_min	Minimum September value of SPEI-6 in the 10-year period between dbh measurements		−1.87	0.12	−2.38	−1.43	x
	SPEI6Sept_avg	Average September value of SPEI-6 in the 10-year period between dbh measurements		−0.11	0.16	−0.76	0.32	x
	SPEI6_ duration	Maximum number of months with SPEI-6 continuously ≤ −1.5 in the 10-year period between dbh measurements	month	5.4	1.1	3	8	x
	SPEI6Sept_share	Share of years in the 10-year period between dbh measurements with September value of SPEI-6 ≤ −1.5	%	14.4	5.1	0	30	x
	PCP	Mean annual precipitation (mm)	mm	1686.5	406.3	850	3600	mc

* BeechT—colline and submontane, montane, altimontane and subalpine, thermophilus, and acidophilus; SoilT—eutric and chromic cambisol, dystric cambisol, leptosol, and other soil types.

Table 2. Description of explanatory variables included in the diameter increment modeling procedure; model: x means the variable was included in the model calculation, and mc means the variable was excluded from the modeling procedure to avoid multicollinearity.

Type	Acronym	Description	Unit	Mean	SD	Min	Max	Model
Dependent	id	Diameter increment at breast height	cm	2.9	2.1	0	14.5
Tree	dbh	Diameter at breast height	cm	26.8	11.8	10	88	x
	dbh/QMD	Quotient dbh/QMD						mc
Competition	BAL	Overtopping basal area	m2 ha−1	18.49	12.06	0	89.22	x
	BA	Stand basal area	m2 ha−1	32.02	10.53	0.87	90.54	x
	CUT	Relative basal area of cut and dead trees	%	10.1%	13.5%	0.0%	99.4%	x
Stand	N	Number of trees per hectare		698	381	40	3000	mc
	QMD	Quadratic mean diameter	cm	26.2	7.2	10.0	68.8	mc
	DDOM	Dominant diameter as mean diameter of the 100 thickest trees per ha	cm	38.6	8.7	10.0	75.3	x
	DMAX	Maximum plot diameter	cm	44.7	10.4	10.0	88	mc
	GINI	Gini index (heterogeneity)	%	0.26	0.05	0.00	0.33	x
Site	ASP	Aspect	1-warmer aspects S, SE, SW, W; 0-colder aspects N, NE, NW, E					x
	SLP	Slope	°	20.1	9.7	0	54	x
	ELE	Elevation	m a.s.l.	688	294.5	80	1629	x
	SProd	Coefficient K as a proxy for site productivity	m3	1.97	0.26	1.20	2.95	x
	DSoil	Depth of soil (cm)	cm	59.3	24.5	0	365	x
	DSoil_A	Depth of soil A horizon	cm	13.5	5.1	0	57	x
	pH	Soil pH (average)		5.12	1.11	0.00	7.60	x
	pH_A	Soil pH of the A horizon		5.07	1.33	0.00	7.50	x
	ORG	Sum of organic matter	%	8.3	3.3	0	38	x
	ORG_A	Sum of organic matter in A horizon	%	3.9	0.9	0	8	x
	BeechT	Beech forest type	Categorical variable *					x
	SoilT	Soil type	Categorical variable *					x
Climate	T	Average annual temperature	°C	8.0	1.5	0	11	mc
	Tmax	Maximum annual temperature	°C	12.8	2.1	7.0	18.5	mc
	Tmin	Minimum annual temperature	°C	3.7	1.5	−1.0	7.0	mc
	Tmax_Jul	Maximum temperature in July	°C	23.0	2.3	16.0	28.0	mc
	Tmin_Jan	Minimum temperature in January	°C	−4.2	1.5	−9.5	1.5	mc
	Tspr	Average temperature for March, April, and May	°C	7.5	2.0	1.0	12.0	x
	Tspr_sum	Sum temperature for March, April, and May	°C	22.4	5.9	3.0	36.0	mc
	SOLRAD	Solar radiation	kJ m−2	1907.5	105.6	1580	2395	mc
	BIO2	Mean diurnal range (Tmax–Tmin)	°C	9.1	2.4	0.0	16.0	mc
	BIO10	Mean temperature of warmest quarter (°C)	°C	16.4	2.0	10.3	21.8	mc
	BIO11	Mean temperature of coldest quarter (°C)	°C	−0.6	1.3	−4.3	5	mc
	SPEI6Sept_min	Minimum September value of SPEI-6 in the 10-year period between dbh measurements		−1.87	0.12	−2.38	−1.43	x
	SPEI6Sept_avg	Average September value of SPEI-6 in the 10-year period between dbh measurements		−0.11	0.16	−0.76	0.32	x
	SPEI6_ duration	Maximum number of months with SPEI-6 continuously ≤ −1.5 in the 10-year period between dbh measurements	month	5.4	1.1	3	8	x
	SPEI6Sept_share	Share of years in the 10-year period between dbh measurements with September value of SPEI-6 ≤ −1.5	%	14.4	5.1	0	30	x
	PCP	Mean annual precipitation (mm)	mm	1686.5	406.3	850	3600	mc

* BeechT—colline and submontane, montane, altimontane and subalpine, thermophilus, and acidophilus; SoilT—eutric and chromic cambisol, dystric cambisol, leptosol, and other soil types.

2.3. Model Formulation

To avoid multicollinearity, we eliminated one of the explanatory variables in pairs for which the Pearson correlation coefficient was greater than 0.8, using the cor() function in the stats R package, v4.1.3 [66]. We eliminated fifteen potential explanatory variables, eleven of them being climate variables (Table 2, marked with mc).

As our data are arranged in a hierarchical structure, the most appropriate method to model the diameter growth of an individual tree is linear or nonlinear mixed-effects modeling. We estimated the linear mixed-effects model with natural splines using the lmer() function from the lme4 package [67]. This formulation incorporated a random intercept with a fixed mean, allowing intercepts to vary among plots while keeping slope variability constant across them. However, when the linear mixed-effects modeling procedure was applied, the variance component associated with the plot-level random effect was small (τ₀₀ = 0.109, p < 0.01), suggesting that plot-specific differences exert only a limited influence on diameter increment. The intraclass correlation coefficient, representing the proportion of total variance attributable to plot-level heterogeneity, was around 0.27, which fell below the commonly cited threshold of 0.5 for supporting the use of a random-intercept mixed model [68]. In addition, parameter estimates from the mixed-effects model were only marginally different from those of the fixed-effects model. Based on these results, we omitted this methodology and proceeded with a generalized linear modeling approach. Our basic model was therefore specified as in Equation (1):

$$\sqrt{\mathrm{i}\mathrm{d}}={\beta }_0+{\sum }_i^n{\beta }_i\ast x_i+{\epsilon }_i$$

(1)

where id represents the 10-year diameter increment of beech, β₀ is the intercept, β_i are coefficients associated with the predictor variables x_i, and ε_i is a residual error term. A single model was developed for all beech trees, and separate models were developed for each mixture, since our focus was the investigation of growth patterns of beech across different mixtures.

To detect potential nonlinear relationships between the response and explanatory variables, id was modeled using natural cubic splines, implemented with the ns() function from the splines R package [66]. Natural cubic splines constitute a flexible extension of generalized linear modeling, approximating nonlinear patterns through piecewise cubic polynomials while enforcing linearity beyond the boundary knots. This formulation generally yields more stable behavior at the extremities of the data range compared with standard cubic splines and prevents unrealistic responses beyond the boundary knots [69]. The model with natural splines is specified as in Equation (2):

$$\sqrt{\mathrm{i}\mathrm{d}}={\beta }_{0i}+{\beta }_{1i}\ast x_i+{{\beta }_{2i}\ast {\left(x_i-a_{1i}\right)}^3+{\beta }_{3i}\ast {\left(x_i-a_{2i}\right)}^3+\cdots +{\beta }_{ki+1}\ast {\left(x_i-a_{ki}\right)}^3+\epsilon }_i\hspace{1em}\hspace{1em}i=1,\dots ,n$$

(2)

where id represents the 10-year diameter increment of beech, β₀ is the intercept, β_i are coefficients associated with the predictor variable x_i, a_i are knots, and ε_i is a residual error term. The model parameters were estimated using the maximum likelihood estimation (MLE) [70]. Splines with one to three knots were generated for variables exhibiting a significant deviation between the residual and component lines and were subsequently incorporated into a stepwise model selection procedure. This was followed by backward elimination, wherein the number of knots in variables with non-significant spline segments was reduced and the model was refitted. Variable inclusion or removal was guided by changes in the Akaike Information Criterion (AIC) computed using the stepAIC() function from the MASS R package, v7.3-56 [71].

When the model for each mixture was developed, we additionally calculated the variance inflation factors (VIF) using the vif() function in the car R package, v3.0-12 [72]. If VIF > 10, the predictor was removed from the model to avoid multicollinearity (Table 2, marked with mc). Since our main focus was the effects of BA and BAL on beech growth, in some models, these predictors were retained despite VIF values being slightly over 10.

After the model was built, extreme outliers were removed within each model with outlierTest() and influencePlot() from the car package, v3.0-12 [72], and the model was refitted with cleaned data. Because the sampling grid density of PSPs varied among regions, plot data were weighted by relative spatial resolution to account for differences in sampling intensity. Relative spatial resolution was calculated as the quotient between the spatial resolution of the observed plot and the mean spatial resolution of all plots.

The predictive performance of the fitted model was assessed using 10-fold blocked cross-validation, implemented with the trainControl() and groupKFold() functions from the caret R package, v6.0-91 [73]. The fit of each model was quantified using the coefficient of determination (R²), root mean square error (RMSE), and mean absolute error (MAE).

3. Results

3.1. Diameter Growth of Beech in Pure Stands and Two-Species Mixtures

The average id of beech significantly differed between pure stands and mixtures (ANCOVA, p < 0.05). In pure stands, the estimated marginal mean id was the second lowest (2.42 cm 10y⁻¹); the lowest was found in the beech–spruce and fir mixture with 60%–90% beech (SPFI1, 2.31 cm 10y⁻¹). The highest diameter growth was observed in the mixture with pine and a proportion of beech up to 30% (PIN3, 3.84 cm 10y⁻¹). Within mixtures composed of the same admixed species, the same pattern was observed across all mixture classes: the estimated marginal mean id increased as the proportion of admixed species increased (Figure 2). The largest difference between mixtures with the same species was found for fir and spruce and fir, for which the highest estimated marginal mean id in fir- (FIR3) and spruce and fir-dominated mixtures (SPFI3) was 45% and 49% higher than in the corresponding mixtures with 60%–90% beech (FIR1, SPFI1), respectively. The lowest difference was found for mixtures with oak (26%).

3.2. Diameter Increment Model for All Beech Trees Irrespective of Mixture

The model for all beech trees, regardless of mixture, explained 26.4% of the variance in the diameter increment of beech. The model is shown in Equation (3), and the coefficients and other parameters are given in Table S3:

√id = β₀ + β₁ ns(dbh,2) + β2 ns(BAL,2) + β3 ns(BA,3) + β4 (CUT,2) + β₅ ns(D_DOM, 3) + β₆ GINI + β₇SProd + β₈ASP + β9 ns(SLP,3) + β10 ns(ELE,3) + β11 ns(DSoil,2) + β₁₂DSoil_A + β₁₃Orgs + β₁₄Orgs_A + β₁₅SoilT + β₁₆BeechT + β₁₇Tspr + β₁₈SPEI6Sept_min + β₁₉SPEI6Sept_avg + β₂₀SPEI6_duration + e_i

(3)

It achieved an RMSE of 0.606 cm and an MAE of 0.471 cm. The most influential predictors were those describing the tree characteristics (dbh) and competition (BA, BAL, and CUT), which together contributed 78.6% to the explained variance. Tree dbh accounted for 36.2% of the explained variability, and competition predictors explained 42.4%, of which CUT, BA, and BAL contributed 19.4%, 16.8%, and 6.2%, respectively. Stand, site predictors, and climate predictors added 4.9%, 12.8%, and 3.5%, respectively.

3.3. Diameter Increment Models for Beech in Pure Stands and Mixtures

There were obvious differences in the predictors included in the diameter increment models among mixtures, as well as in the magnitude and direction of their effects (Tables S4–S6). Model performance differed among mixtures as well, as R² ranged from 20 to 34% in most mixtures, with the lowest observed in mixtures with oak and pine (

Table 3. Performance of the models across different mixtures and in pure beech stands; R² is the coefficient of determination, RMSE is the root mean square error, and MAE is the mean absolute error.

Mixture	R2 (%)	RMSE (cm)	MAE (cm)
OAK1	28.5	0.619	0.481
OAK2	24.4	0.632	0.494
OAK3	26.4	0.624	0.492
MAP1	31.2	0.557	0.430
MAP2	33.8	0.577	0.457
MAP3	67.9	0.494	0.409
PIN1	29.1	0.599	0.478
PIN2	19.5	0.646	0.519
PIN3	21.6	0.634	0.498
SPR1	31.1	0.575	0.446
SPR2	31.2	0.595	0.460
SPR3	25.3	0.606	0.469
FIR1	33.0	0.514	0.396
FIR2	32.5	0.505	0.390
FIR3	29.8	0.550	0.430
SPFI 1	29.7	0.537	0.415
SPFI 2	30.8	0.540	0.422
SPFI3	29.9	0.554	0.427
BEE	29.3	0.566	0.438

). An exception with much higher R² was the model MAP3, which was, however, developed on a small sample of 70 trees; thus, the results should be interpreted with caution. When validated, RMSE ranged from 0.494 to 0.646 cm, and MAE ranged from 0.390 to 0.519 cm; again, mixtures with pine and oak showed the highest prediction errors (Table 3).

Across mixtures, the most important predictor was tree dbh, which was included in all models except in maple- and pine-dominated mixtures (MAP3 and PIN3) (

Table 4. The relative contribution of predictors in the models of beech diameter increment in mixtures and pure beech stands; the relative contribution is expressed as the relative decrease in R² (%) when the predictor was omitted from the model and the model was refitted; at the bottom, the summed relative contribution to the explained variability by groups of variables is shown.

Mixture	OAK1	OAK2	OAK3	MAP1	MAP2	MAP3	PIN1	PIN2	PIN3	SPR1	SPR2	SPR3	FIR1	FIR2	FIR3	SPFI1	SPFI2	SPFI3	BEE
dbh	51.6	31.5	45.1	45.1	41.5		16.0	67.2		37.7	46.9	24.8	49.1	52.2	41.8	39.8	48.8	35.2	41.3
BAL	6.9	8.7		6.7	1.4	74.6	43.9		65.0	4.9	13.1	1.7	4.7	1.7		10.0	4.7	5.1	6.0
BA	14.0	17.6	37.2	18.5	23.1			10.6		28.0	7.8	9.3	23.2	16.0	36.1	14.2	3.3	2.5	21.4
CUT	11.3	21.6	8.2	14.5	6.4		24.9	6.3		17.8	11.4	27.1	4.9	11.8	10.8	4.4	8.4	20.1	13.4
DDOM			2.4	3.5		7.3			12.5	1.2						10.6	1.3	9.4	7.8
GINI	2.2				5.3					0.6			0.7	0.6	1.1		2.3		1.4
Sprod	0.3		3.1				3.6		12.7			2.0						4.0	0.3
ASP		5.5		0.4		8.2					0.9	1.2	0.7					2.3	0.2
ELE				0.5			2.7			1.3	2.2	17.4	11.5	6.3	2.8	11.0	15.6	9.3
SLP	1.4										2.5			4.8	2.0	2.8	6.8		0.5
DSoil	0.7									0.8	0.7			1.3				1.8
DSoil_A	0.8	2.8		0.6						0.3	0.8	1.1					0.6		0.2
Orgs				0.4			4.6	2.1	9.8		1.2	1.1		0.5				2.8
Orgs_A	1.8									1.1	3.0						1.3
SoilT	1.4	2.6	4.1	0.5	3.0		2.2	6.0			1.3	3.6	1.1	2.9		1.7	1.2		0.4
BeechT	2.0	4.2		8.2	5.7	9.9				1.9	2.6	1.4			4.5	4.7	3.1		1.9
Tspr				1.1			2.2					1.9					2.5	5.9	0.5
SPEI6Sept_min	3.8				2.2					0.4			2.8	2.0					2.6
SPEI6Sept_avg	0.5	2.4						7.7		0.5	1.0	3.4	0.8		0.9				1.7
SPEI6_duration					6.7					0.5	1.9	0.8	0.6			1.0		1.6	0.2
SPEI6Sept_share	1.2	3.1			5.9					3.1	2.7	3.2							0.2
Tree effects	51.6	31.5	45.1	45.1	41.5	0	16.0	67.2	0	37.7	46.9	24.8	49.1	52.2	41.8	39.8	48.8	35.2	41.3
Competition effects	32.2	47.9	45.4	39.7	30.9	74.6	68.8	16.9	65.0	50.7	32.3	38.1	32.8	29.5	46.9	28.6	16.4	27.7	40.8
Stand effect	2.2	0	2.4	3.5	5.3	7.3	0	0	12.5	1.8	0	0	0.7	0.6	1.1	10.6	3.6	9.4	9.2
Site effect	8.4	15.1	7.2	10.6	8.7	18.1	13.1	8.1	22.5	5.4	15.2	27.8	13.3	15.8	9.3	20.2	28.6	20.2	3.5
Climate effects	5.5	5.5	0	1.1	14.8	0	2.2	7.7	0	4.5	5.6	9.3	4.2	2.0	0.9	1.0	2.5	7.5	5.2

). The relationship between id and dbh differed between mixtures, as natural splines with 0–2 knots were determined, reflecting different relationship patterns. The decline in beech id with increasing dbh was most pronounced in stands where beech was admixed with spruce and fir. We detected a linear relationship in mixtures with admixed pine (PIN1, PIN2) and oak (OAK1, OAK2), but also in spruce (SPR3) and fir-dominated stands (FIR3). In two mixtures with fir, the increment curve showed an increase with increasing dbh at lower values, then remained static for mid-sized trees, and subsequently it began to rise again for large-sized trees (Figure 3). A logarithmic-shaped curve with a decline at larger dbh, representing the expected growth pattern, was observed only in pure beech stands and in stands with lower admixture of oak and spruce (OAK1, SPR1, and SPR2).

Predictors describing competition were strongly influential in all models, accounting for 16%–75% of the explained variability, with an average of 40% (Table 4). There was no consistent pattern specifying whether BA, indicating size-symmetric competition, or BAL, indicating size-asymmetric competition, was more important within a specific mixture; however, in general, BA appeared to have a larger influence (Table 4). BA appeared more important in pure beech stands and in several mixtures with oak, spruce, fir, and maple, with the exception of maple-dominated stands (MAP3). With increasing BA, id declined, most often sharply down to approximately 30 m²/ha, while beyond this threshold, the decline became more gradual. Curves with a gentler decline were observed for mixtures with spruce and fir, while the most rapid reduction in id with increasing BA occurred when beech grew in mixtures with maple and pine (Figure 3).

BAL was particularly influential in maple- and pine-dominated stands (MAP3, PIN3), but also in beech stands with a lower proportion of admixed pine (PIN1). In spruce–fir–beech mixtures, BAL was more important than BA in two of those mixtures. The largest reduction in id with increasing BAL occurred when beech grew in mixtures with maple and pine, while in other mixtures, the slope of the curve was gentler and similar across mixtures.

CUT exerted a substantial influence, in some cases surpassing the influence of BAL and BA. CUT appeared to have the largest influence in mixtures with spruce and oak (Table 4). In all models, CUT exhibited a progressively increasing relationship with id.

The effects of other predictors were generally smaller (Table 4, Figure 4 and Figure 5). The influence of D_DOM varied among mixtures and was included in all models containing spruce and fir, where its contribution to the explained variability was the highest. ELE played an important role in mixtures containing spruce and/or fir and was included in all models for mixtures with at least one of these species. For all mixtures where it was included, id increased with ELE and decreased with SLP. In contrast, SProd exhibited an opposing effect across mixtures. In spruce-dominated (SPR3) and spruce and fir-dominated stands (SPFI3), this relationship was surprisingly negative and linear, indicating lower id at higher site productivity. Soil parameters played the greatest role in mixtures where beech is mixed with pine, followed by mixtures with spruce and oak (Table 4); however, the direction of their effects was not always consistent (Figure 4 and Figure 5). Climatic parameters appeared to exert a stronger influence in mixtures with spruce but also stood out in some models for maple (MAP2) and pine (PIN2) (Table 4). For all mixtures, id increased with higher SPEI6Sept_min, indicating better growth in areas with lower drought intensity. Average SPEI6Sept_avg showed a diverse relationship, as it increased id in five mixtures and decreased in four mixtures.

4. Discussion

In the study, we aimed to develop diameter-growth models for beech across mixtures with various tree species, including tree, competition, stand, site, and climate variables, and, in particular, to determine whether the relevance of size-symmetric and size-asymmetric competition differs between mixtures.

The performance of the 19 models we developed ranged from 19% to 34%, despite the large number of variables included in the modeling procedure. The low performance rate of the models is not an issue as the models were mainly intended for studying influential factors and competition mechanisms among mixtures rather than for high-precision prediction of individual tree growth. Nevertheless, it was comparable to that reported in similar studies, e.g., [43,74,75], while [10,29,76] reported substantially higher R² values. The ability to explain beech diameter growth was quite similar among mixtures, although a somewhat poorer model performance was observed in mixtures with pine. The low overall performance suggests that additional, unmeasured factors influence beech diameter growth in these mixtures. The diameter growth of pine and oak, and thus also the competitiveness of both species in mixtures, strongly depends on soil conditions [74]. More detailed soil data at sites where beech co-occurs with these and other species might improve growth models. The performance of models in different mixtures was likely influenced by the number of sample plots and trees as well. In mixtures with fewer plots and trees, the variability in independent predictors was lower, which may have affected the results. In the mixture with maple where beech constitutes <30% (MAP3), the results are likely to be unreliable due to the small sample size [10].

In the final models, the influence of predictors differed among mixtures. Different sets of predictors were included in the models. They explained varying amounts of variability, and their effects differed in magnitude and direction.

4.1. Competition as a Crucial Predictor of Beech Diameter Growth

Competition-related predictors BA, BAL, and CUT were recognized as the most influential, as they explained, on average, 40% of the variability in id. Competition is known to be an important driver of tree growth [9,22,26], and for beech it was often found to be the most influential predictor, far more than site or climatic conditions [24,77]. In mixtures in which competition variables were less important, other predictors, especially site-related predictors, contributed considerably more to the explained variability of id (e.g., in mixtures with spruce and fir). In these mixtures, the negative impact of competition for belowground resources or light seems to be less crucial for beech growth. A probable reason for this may be that, in these mixtures, site conditions represent the dominant limiting resource for beech growth, or even for the growth of all species [24].

There was no clear pattern indicating whether size-asymmetric or size-symmetric competition influenced beech diameter growth more across or within mixtures with a certain tree species. Most models included both components, as was also the case in Switzerland [31]. However, in general, the prevailing mode of competition seemed to be size-symmetric competition, represented by BA. This means that competition for belowground resources, including nutrients and water, seems more important than competition for light. This is in line with Mina et al. [31] but contrasts with the findings of del Río et al. [18]. Size-symmetric competition was found to be more pronounced on low-productivity sites, while size-asymmetric competition was more pronounced on more fertile sites [24].

The BA contribution to the explained id variability was greater than that of BAL in pure beech stands and in mixtures with fir, oak, and maple (the exception was MAP3). This confirmed the importance of the size-symmetric competition and belowground competitive ability of beech [18,78]. If admixed with fir or oak, beech might be assumed to benefit from the hydraulic redistribution of water taken up by deeper rooting systems of the admixed species [79]. In addition, several studies reported beech’s ability to make morphological and physiological adjustments to its rooting system compared to competing species in mixed stands [77,80]. For example, the fine-root development of beech may be facilitated in mixtures with spruce due to reduced intraspecific competitive pressure [80].

In most mixtures, id decreased sharply with BA up to approximately 25–30 m²/ha, while beyond this threshold the decline became more gradual. Compared to beech growth in mixtures in Spain [22], the pattern was similar, but at lower stand densities than the decrease in id, and in our case, much more rapid. This means that size-symmetric competition for belowground resources increases sharply with increasing stand density. At higher stand densities, the negative influence of size-symmetric competition diminishes; however, it is not obvious which predictor becomes more influential at these stand densities. An exception was mixtures with beech–spruce and fir. In all mixtures of these species, id at the lowest BA was lower than in other mixtures, and the overall relationship was divergent, and in two mixtures it was even linear. However, the contribution of BA in these models was lower than that of BAL and CUT.

In contrast, size-asymmetric competition, measured with BAL, appeared to possess greater significance than size-symmetric competition in mixtures dominated by pine (PIN3) and maple (MAP3), with a smaller contribution also relevant in the three-species mixture of beech–spruce and fir (SPFI). Competition for vertical space and light is obviously a crucial driver of beech diameter growth in these mixtures, indicating its lower competitiveness on specific sites. There, the admixed species are more adapted to specific site conditions, such as maple on moist, nutrient-rich soils or Scots pine on shallow, sandy, and dry soils [81]. In mixtures with pine, the results were unexpected. Given that sites of beech–pine mixtures are the least productive (Table S2) and that size-symmetric competition has been identified as the dominant mode of competition for pine diameter growth in Spain [26], we expected a similar pattern in our study. Instead, size-asymmetric competition seems to be more influential than size-symmetric competition. On these specific sites, maple or pine, as light-demanding species, exhibit rapid juvenile growth, enabling them to overtop beech and subject it to increased shading and size-asymmetric competition. This might relate to the crown size of trees. The size and shape of the beech crown differ depending on whether it grows in a pure stand or in a mixture [36]. In addition, the crown size varies among species at the same dbh; for example, maple usually exhibits a larger crown than beech at the same dbh [82,83].

BAL was also an important predictor in the three-species mixture of beech–spruce and fir. This confirmed the finding of Mina et al. [10] that, in these mixtures, beech diameter growth was subjected to competitive reduction if the larger-sized trees were fir or spruce. For beech, the composition of overtopping competitors was found to be important, suggesting that how species are stratified in a stand is crucial for modulating its growth [84]. We did not study these interactions directly, but with a higher proportion of fir and spruce (SPFI2 and SPFI3), the mean diameter increment of beech was higher, and the contribution of BAL to the explained id variability was greater than in SPFI1, with a proportion of beech of 60%–90%. This signals accentuated competitive reduction or even facilitation on beech growth, which is in accordance with [31]. However, it may also happen due to strongly expressed intraspecific competition in mixtures with a higher beech proportion due to the intense lateral expansion of its crowns [36].

The diameter growth of beech clearly benefited from the admixture of other tree species, as estimated marginal means of id for all mixtures except one were higher than that of pure beech stands. This confirms findings of other studies that beech is a strong self-competitor [28,31,36] and that intraspecific competition often exceeds interspecific competition in beech mixtures [18,29,84].

Another variable indirectly expressing competition was CUT, which quantifies the proportion of harvest and mortality in a ten-year period, which was found to also be an important predictor in other studies, especially for broadleaves, e.g., [29,31]. CUT increased diameter growth and was more important than BA and BAL in several models, indicating a strong effect of management on competition between trees and, consequently, their growth. Harvesting increases growing space of individual trees and, consequently, reallocates growth to selected crop trees [85]. Harvest reduces BA and, for the most retained trees, also BAL, although some dominant trees may not experience the latter. In young and middle-aged stands, the way BA and BAL are reduced depends on the thinning method. In Slovenia, selective thinning, or thinning from above, is predominantly applied [85]. This means that harvesting is mainly oriented to help the dominant crop trees by enlarging their growing space. Timber harvest, however, can also be realized for regeneration purposes in mature and uneven-aged stands or as a salvage harvest after natural disturbances. Harvest intensities in thinning range 5%–50% of stand basal area or stand volume, while in other regular, regeneration, and salvage harvests the range is 5%–100%. With higher intensity of harvesting, competition is progressively reduced, resulting in progressively increasing diameter growth of retained trees. As CUT also embraced tree mortality, it captures a comprehensive effect of regular harvesting and natural disturbances on tree growth and development.

4.2. Other Predictors of Beech Diameter Growth

The most important predictor in the majority of mixtures was tree dbh, which can also be classified as a measure of past competition experienced by the tree [42]. As dbh increased up to 20–25 cm, id increased in all mixtures, but after that threshold, differences between mixtures became apparent. The biological dynamics of diameter growth are best emulated by the curve in which id first increases with increasing dbh, reaches the culmination at a certain dbh, and starts to decline afterward, e.g., [42,75,76]. Such a response was expected to be common across mixtures, but it was identified only in pure beech stands and some mixtures with spruce, spruce and fir, and oak. The id of beech declined the most strongly at large tree sizes when admixed with spruce and fir, expressing an almost parabolic relationship. In two mixtures with fir, the beech growth pattern was unexpected. After an increase and reaching the expected maximum at mid-sized dbh, id temporarily stabilized and then increased again, reaching the maximum values at the largest dbh. This may be related to the fact that these plots exhibited the least homogeneous stand structure among the studied forests, and for heterogeneous fir–beech forests, such a pattern was already identified [58]. In addition, in several mixtures with pine and oak, the dbh-id relation was even linear, which is also in accordance with some other studies, e.g., [46].

All other stand, site, and climate factors contributed much less to the explained variability of id, but in some models, individual variables nevertheless contributed a considerable part of the explained variability. Elevation (ELE) was included in models for 11 mixtures; the greatest influence was found in beech mixtures with spruce, fir, or both. The relationship with id was negative and linear in all models, except in SPR1, where natural splines had one knot expressing a progressively declining relationship. The latter was also observed for basal area growth of beech trees on the country-wide level [39], while many studies confirmed the decrease in diameter growth with increasing elevation, e.g., [86]. Related to changing climate conditions, elevation-specific diameter growth changes were detected for beech, and this pattern may be expected to continue in the future, which may not be the case for spruce and fir [21]. SLP was less important and was included in models for only six mixtures. In all these models, the diameter growth of beech constantly dropped with increasing slope, with the exception being SPR2, which had a slight increase between 15° and 30°. On steep slopes, growing conditions, especially regarding water supply, but also nutrient supply, are harsher, which may negatively affect the diameter growth of trees.

The effect of D_DOM was difficult to interpret, as its influence varied substantially among mixtures. In some mixtures, the relationship was parabolic, showing the lowest id at mid-sized D_DOM. Most surprising were its influences in SPFI2, SPR3, and MAP3, in which the diameter growth of beech was negatively and linearly related to D_DOM. The underlying reasons for this pattern remain unclear. Related to the Gini index, beech exhibited higher diameter growth in less homogeneous stands across all mixtures except one (FIR1). This probably indicates more efficient light and belowground resource use in structurally diverse stands, although this was not tested directly.

Soil variables had the strongest influence in beech mixtures with pine and slightly less in spruce and maple mixtures, all likely indicating the specificity of such sites. Specific soil conditions in sites where beech is admixed with or to pine and maple may confer an advantage to the admixed species. Pine thrives in sandy nutrient-poor soil with low water-holding capacity [81], which is too extreme for vigorous growth of beech. In contrast, maple thrives in nutrient-rich and moist soil [81], which is also proper for beech, but as maple is a light-demanding species, its dominance on such sites is also related to a rapid height growth of trees in early stand stages. Related to soil and belowground relationships between trees, possible distinguishing factors of beech growth in mixtures might also be diverse mycorrhizal relationships in pure beech forests and in various mixtures. Also, allelopathy or exposure to pollution might be important factors influencing individual tree diameter growth as both decrease the vitality of trees of a certain tree species; however, they were not addressed in our study. Site productivity was recognized as a predictor in only six models, and in some of them it appeared to have an unexpected effect on id. Diameter growth increased with increased Sprod, as was expected in pure beech stands and in mixtures with pine and oak (PIN1, PIN3, OAK1, and OAK3), while in mixtures with dominant spruce (SPR3) and spruce and fir (SPFI3), SProd unexpectedly had a negative linear effect. In confirmation of the absence of SProd in several models, Pretzsch and Biber [24] found that productivity did not improve id estimation of beech, while it was an important predictor for other species. A plausible explanation seems to be that higher SProd enhances beech growth but simultaneously increases the growth of competing species, and increased competition reduces beech id. In addition to id growth, SProd also influences stand density [9] and tree height [39], and thus spruce and fir may become more competitive on more productive sites as they reach larger dimensions than beech.

For climate predictor influence, there was no clear pattern across mixtures. Mean spring temperature (Tspr) increased id in four mixtures. At high elevations, beech growth was often recognized as being temperature-driven, whereas at low elevations it was predominantly influenced by summer rainfall and negatively affected by high summer temperatures, e.g., [38,51]. Tspr reduced id in PIN1, probably due to spring drought stress related to low water-holding capacity of soil at these sites. Beech was shown to be highly sensitive to warm and dry periods [29]. With decreasing drought stress, measured by SPEI6Sept_min, id increased as expected in all mixtures where it was included in the model. SPEI6Sept_avg had varying effects across mixtures. If the index was higher, meaning less dry conditions, id increased in four mixtures with pine, spruce, and oak. The opposite occurred in three mixtures where beech mixed with fir or oak and in pure beech stands. This implies that such metrics of drought (in addition to SPEI6Sept_avg, also SPEI6_duration and SPEI6Sept_share) are not the most suitable metrics to be used in forest ecology modeling where different approaches are required than in agriculture. In the average SPEI value, severe drought can be compensated by very wet years, which can obscure the modeling results. Drought metrics emphasizing extremes would probably be more appropriate than mean values [43]. Our results suggested the variable influence of climate and drought on beech diameter growth across mixtures; thus, more research is needed in this area.

4.3. Methodological Aspects and Limitations

The surrogates for size-symmetric and size-asymmetric competition were BA and BAL, indices that were not calculated from crown characteristics and spatially explicit data on individual trees. Indices based on this information were found to be superior to distance-independent indices such as BA and BAL [87] because crown morphology differs among tree species [5], but they were not superior in stable stand structures in unmanaged stands [88]. Data gathered in regular forest management and national forest inventories usually do not enable calculation of crown-characteristics-based indices because insufficient data are collected there [18,62]; thus, spatially non-explicit indices are more easily applicable.

A further limitation of our study is the use of 10-year diameter increment data coupled with explanatory variables measured at the beginning of the inventory period. Although this is a common approach in growth studies based on national forest inventory data, e.g., [43], it is important to recognize potential changes in stand structure between two measurements. In our analysis, these potential changes were considered only through the CUT variable, while all other explanatory variables described conditions at the beginning of the inventory period.

The climate variables included were mainly long-term averages, which did not directly address extreme events during this period that can be crucial for beech growth, e.g., [43]. However, we included the SPEI to detect the influence of severe or long-term droughts, but for the last three years of the observation period, we could not calculate SPEI due to a lack of data. In 2013, a significant drought was observed in Slovenia [89]; therefore, the absence of these data could represent a potential bias in our results.

Another aspect to note is how mixtures were defined. We developed models for pure beech stands and various two- and three-species mixtures of beech with other species. Species composition and species proportions were thus captured through mixture category. Some other studies investigated the effects of mixture on tree growth by including species-specific competition indices (e.g., BA or BAL per species) in the modeling procedure [15,22,26,31]. This approach was also tested in our study, but correlations between species-specific BA and BAL were high; thus, we chose a different approach. In addition, we consider our approach to be more management oriented as field foresters often classify forest stands into categories of species composition.

It is difficult, however, to interpret the effect of mixture on diameter growth without considering the site where each mixture typically occurs [9]. In our study, some results most likely reflected site effects associated with specific mixtures rather than the intrinsic effects of mixture on beech growth. Site was represented by multiple explanatory variables; however, results would likely be more robust if the study was conducted on a single site type where mixtures with all species in different proportions occur.

Data from regular forest management inventories and national forest inventories [62] usually cover large forest areas, and the same was true of our database. In addition, the dataset encompassed a wide range of stand characteristics and wide ecological gradients of all relevant site conditions. These characteristics of our dataset offset the limitations described above.

5. Conclusions

Our study confirmed that modeling diameter growth of individual trees using only explanatory variables gathered in regular forest inventories (or calculated from them) is a useful tool to explore which predictors have the greatest influence and whether size-symmetric or size-asymmetric competition is more important for beech diameter growth in pure forests and two- and three-species mixtures. In general, size-symmetric competition appeared to have a larger influence across all mixtures, but size-asymmetric competition was a crucial competition predictor in some mixtures with pine, maple, and spruce and fir. Both can be influenced by forest management through affecting stand density and competition for belowground resources (by influencing BA) or through altering vertical structure and thus competition for light (by influencing BAL). Beech was confirmed to be a strong self-competitor, and intraspecific competition appears to have a greater impact on its growth than interspecific competition, as it grew much better in mixtures with various species than in pure forests. Niche complementarity between admixed species, competitive reduction, and facilitative effects on beech in mixtures resulted in increased diameter growth. Therefore, we recommend growing beech in mixtures with other species and applying forest management approaches that account for the symmetry of competition, which was found to be crucial in each two-species or three-species mixture.