Sustainable regeneration in uneven-aged mixed deciduous forests managed by selection silviculture: the role of demographic structure

Selection silviculture aims to create and maintain uneven-aged forests with a diameter at breast height (DBH) structure that is balanced at small spatial scales such that the stem number in each DBH class is high enough to replace the harvested and outgrowing trees over time by ingrowth from the next lower class. In these forests, natural regeneration of shade-tolerant species is at an advantage over shade-intolerant species. This is particularly pronounced in continuous-cover stands dominated by European beech ( Fagus sylvatica L. ), which develops its crown laterally as a reaction to release events. The conditions necessary to sustain a mixture with less shade-tolerant species, e.g. sycamore ( Acer pseudoplatanus L .), have been little studied. Therefore, we explored growth patterns and stand structures in mixed deciduous forests with light availability quantiﬁed using vegetation height models. Harvesting and growth patterns were derived from inventory data of beech-dominated selection forests in Thuringia, Germany and long-term forest monitoring plot data from four stands in Switzerland. Based on these data, models of stem number distributions conﬁrmed that stand basal area in deciduous forests dominated by beech should not exceed 21–25 m 2 ha − 1 to maintain a sustainable structure. In these forests, a total of ∼ 90–120 stems per ha are needed in the ingrowth DBH class (DBH 8–11.9 cm) to ensure demographic sustainability. At canopy light transmittance < 10 per cent, total stem number required in the thicket stage (DBH < 8 cm) is 800–1700 ha − 1 . Under such shady conditions, only a small proportion of sycamore was observed ( < 25 per cent) in the thicket stage, as the species is likely to require canopy gaps > 400 m 2 (gap diameter > 22.5 m) to recruit successfully. Selection silviculture with shade-intolerant species therefore requires much lower stocking volume and larger canopy openings created by group selection cutting than what is routinely applied in practical forest management using single-tree selection principles


Introduction
The sustainable provisioning of forest ecosystem services (ES) is a major goal of forestry.Different silvicultural systems have emerged to maintain specific forest structures to maximize certain ES, such as protection against natural hazards, recreational value or timber supply, while others are used for multifunctional objectives to promote a multitude of ES simultaneously (Schütz, 1999;Brang et al., 2014).A prominent example of silvicultural systems that simultaneously provide multiple ES is continuouscover forestry (CCF), where the maintenance of a permanent canopy cover is sought (Pukkala and von Gadow, 2012).Interest in this system has increased tremendously in central Europe during the last decades (Pommerening and Murphy, 2004).
Within the framework of CCF, selection silviculture specifically aims to create and maintain uneven-aged forest structures by the regular cutting of single trees or small groups of trees (Nyland, 2016).Regeneration, tending and harvesting take place in the same intervention, and each harvest typically provides some timber revenue.To ensure the permanent and long-term provisioning of ES, different concepts for controlling the stocking were developed (O'Hara and Gersonde, 2004;O'Hara et al., 2007).An early concept was the so-called 'check method' (in French: 'méthode du contrôle') dating back to 1878, which aims to optimize wood production by repeatedly assessing and evaluating stem number frequencies by diameter at breast height (DBH) classes.These are then used to adapt the subsequent silvicultural intervention (Biolley, 1901(Biolley, , 1887)).This foundation of adaptive ecosystem management was further developed by explicitly quantifying the stem numbers based on negative exponential distributions (De Liocourt, 1898;Meyer, 1952;Kerr, 2014).
However, negative exponential functions alone do not reveal conditions of demographic sustainability, as the decay rate may not be constant over DBH (O'Hara, 1998;Rubin et al., 2006).To maintain a sustainable distribution, there needs to be sufficient trees recruited into each DBH class to compensate for mortality, Downloaded from https://academic.oup.com/forestry/article/95/2/201/6371349 by Lib4RI -Library of Eawag, Empa, PSI, WSL user on 13 April 2022 Forestry harvesting and outgrowth, i.e. when a tree grows from one size class into the next larger size class (Schütz, 2006).If the latter condition is met for the entire distribution, the stand is in a demographic equilibrium that depends on stand basal area (BA), climatic variability or other factors that influence the success of regeneration and the growth rates of recruitment via the light transmitted to the respective canopy layers.It is more important to identify the conditions of this demographic sustainability than to identify any single component, such as the best function describing a hypothesized equilibrium structure (Schütz, 2006).
Single-tree selection silviculture with deciduous species has proven to work well with European beech (Fagus sylvatica L.) at a target stand BA of 22-24 m 2 ha −1 (Schütz, 2006;Hessenmöller et al., 2018).However, it remains unclear under what conditions the selection system may be suitable for maintaining mixtures of tree species with contrasting shade tolerance within the optimum of beech, i.e. the recruitment of less shade-tolerant tree species in the desired amount and quality.Such knowledge is needed for managing mixed deciduous forests towards higher tree species richness, which is important for enhancing the adaptation potential of forests in the face of ongoing global changes (Brang et al., 2014).
In central Europe, beech is the naturally dominant tree species on mesic sites up to an elevation of ∼900-1000 m a.s.l.It is known for its high shade tolerance and competitiveness (Packham et al., 2012).In beech-dominated forests, management interventions directly determine tree species diversity by the intensity of the initial canopy opening (Barna and Bosela, 2015).In single-tree selection silviculture, typically only small canopy gaps in the extent of one or two mature tree crowns are opened and thus often only beech is able to establish and survive as advance regeneration (Madsen and Hahn, 2008).Although juvenile beech trees develop plagiotropic branches when suppressed (Packham et al., 2012), the species retains a remarkable ability to react to release events (Trotsiuk et al., 2012).Unlike coniferous species, upperstorey individuals are able to access newly available canopy space by quickly expanding lateral branches (Emborg, 1998), thus closing the window of opportunity for the regeneration of gap specialists such as sycamore (Acer pseudoplatanus L.).
Less shade-tolerant tree species are therefore often outcompeted by beech (e.g.Petriţan et al., 2007;Ligot et al., 2013), which renders the management of such mixtures using selection silviculture a major challenge.There is evidence for a decline of tree species richness in single-tree selection forests (e.g.Klopcic and Boncina, 2012;Barna and Bosela, 2015;Nolet et al., 2018;Schall et al., 2018;Brüllhardt et al., 2020b), thus questioning the appropriateness of the system for maximizing the provisioning of multiple ES that depend on tree species richness, or for increasing forest resilience that partly depends on tree species richness as well (Zhang et al., 2012;Silva Pedro et al., 2015).As many questions remain about the influence of single-tree selection silviculture on the forest structure, stand density and BA that warrant a certain tree species diversity (O'Hara et al., 2007), it is important to evaluate the growth rates of less shade-tolerant species in the typical light environments of uneven-aged mixed deciduous forest stands.
In the specific case of the Swiss Plateau, single-tree selection silviculture is inspired by the classical systems applied to upper montane coniferous forests characterized by very low light availability in the understorey (Schütz, 2001a).Often, canopy gap size is in the range of a single tree crown, and stand BA may be too high to provide recruitment rates that are sufficient to maintain the stand in demographic equilibrium (Schütz, 2006).Clearly, light availability reaching the trees in the lower diameter classes needs to be high enough to ensure a diameter increment corresponding to sustainable ingrowth rates.To date, however, the quantification of such light levels in mixed deciduous forests is lacking.
Therefore, we aim to explore the conditions that are necessary to sustain a demographic equilibrium with the presence of two contrasting species with respect to their shade tolerance, so as to gain insights into the possibilities and limitations of admixing light-demanding species in selection forests dominated by beech.The central research questions for our study thus were: a.What diameter growth rate corresponds to a sustainable stem number ingrowth rate of sycamore and beech?b.What light availability is necessary to attain these ingrowth rates?c.What minimal gap size ensures the ingrowth of sycamore?
To answer these questions, we assessed annual diameter increment of juvenile beech and sycamore in regeneration cohorts typically found in selection forests.Combined with harvesting patterns of beech selection forests in Thuringia, Germany and growth patterns from long-term monitoring plots in Switzerland, models of stem number frequencies at demographic equilibrium were derived to evaluate sustainable recruitment conditions.

Study area
The study focused on beech forest types across the Swiss lowlands and in the Hainich region of Thuringia, Germany, which are representative for a wide range of central European forests (Figure 1).The elevation of the studied forests ranged from 300 to 860 m a.s.l.mean annual precipitation was 890-1300 mm at the Swiss sites and 600-750 mm in Thuringia.Mean annual temperature was 8 • C to 11 • C in the Swiss lowlands and 6-8 • C in Thuringia.
Soils at the study sites in the Swiss lowlands are slightly acidic to highly alkaline, and mesic to be classified mainly as Luvisols or Cambisols.Soil types at the Thuringia study sites are dominated by a loess cover on a limestone bedrock characterized by a high loam-clay content, resulting in a high water-holding capacity classified as Luvisols, Stagnosols and Cambisosl (cf.Hessenmöller et al., 2018).

Datasets
We used several datasets from three sources to study unevenaged forests with natural regeneration initiated by a periodic selection cutting of individuals or small groups of trees.Stem disc data to reconstruct single tree growth patterns were available from the extensive field campaign presented in Brüllhardt et al. (2020b).Growth data were available from four long-term monitoring plots of mixed deciduous forests within the experimental forest project (Forrester et al., 2019) and used to model stem number distributions to study demographic sustainability in Forestry uneven-aged mixed deciduous forests.These were further combined with harvesting patterns from a large repeated inventory of a balanced uneven-aged beech forest in Thuringia.

Growth rates from stem disc analysis
We selected 14 study sites across the Swiss lowlands where single-tree selection cutting is applied, typically with a frequency of 5-10 years.Regeneration cohorts in small gaps (n = 240) with different sizes of canopy openings were selected on site types of the sub-alliance Galio-odorati-Fagenion (mull beech forests) and Luzulo-Fagenion (moder beech forest) (cf.Leuschner and Ellenberg, 2017).Within each cohort, dominant or co-dominant individuals of sycamore (n = 178) and beech (n = 196) were harvested between 2015 and 2018 for stem disc analysis to reconstruct annual growth rates based on tree-ring analysis.Annual ingrowth rates per DBH class were calculated using the average diameter increment of the last five years divided by the width of the respective DBH class.
The sample trees covered a DBH range of 1-19 cm growing in light environments of 2-60 per cent of above-canopy radiation.Gap size was defined as the extended gap by the two distances from the stem of canopy trees in the south to the north and from the west to the east.Around each of the sampled trees, circular plots of 2 and 15 m radius were set up to record all tree species of DBH < 8 cm and DBH > 8 cm, respectively.
Light availability was estimated for each of the sample trees using a novel approach mimicking hemispherical photographs derived from vegetation height models (VHM).We took highly overlapping aerial photographs of each study site to assess forest geometry photogrammetrically as a 3D point cloud using structure-from-motion (SfM) and multi-view-stereo (MVS) processing.Plotting VHM point clouds in polar coordinates results in a synthetic hemispherical picture (Moeser et al., 2014;Zellweger et al., 2019;Brüllhardt et al., 2020a), which was analysed using Hemisfer 2.2 (Schleppi et al., 2007;Thimonier et al., 2010) to estimate light transmission, plant area index, canopy openness and canopy coverage.As a measure of light availability, we used the gap light index (GLI), which is defined as the average transmission of direct and diffuse light during the growing period.Canopy cover was calculated by subtracting the total gap fraction from the theoretical maximum cover of 100 per cent.A complete description of the workflow can be found in Brüllhardt et al. (2020a).

Experimental forest management data
The experimental forest management (EFM) project maintains a network of permanent monitoring plots across Switzerland (Forrester et al., 2019).We selected four plots from that network in mixed deciduous uneven-aged forests of the Swiss lowlands that are managed by selection silviculture and are located in the same stands as those sampled for the stem disc analysis.The plots were located at Deinikon (plot number 2050000), Adlisberg (3025000), Basadingen (2048000) and Muttenz (3021000).These EFM plots are between 1.3 and 2.5 ha in size and are measured every 5 or 6 years according to the local cutting cycle.The plots were established in (respectively) 1998, 1999, 2003 and 2004, and all trees with a DBH > 8 cm are measured.

Inventory data of uneven-aged beech forests in Thuringia
In north-western Thuringia, Germany, selection cutting in cooperative community forests has a long tradition that led to unevenaged forests dominated by beech.In the forest of the cooperative Hainich, permanent concentric inventory plots were established on a 1 ha grid in 1994 and reassessed in 2004 and 2014 (cf.Hessenmöller et al., 2018).In total, we analysed 1118 plots of 500 m 2 where all trees with a DBH > 30 cm were measured.Smaller trees were assessed on subplots of 200 m 2 (DBH 20-29 cm) and 100 m 2 (DBH 7-19 cm).

Ingrowth and harvesting rates
The repeated measurements of the EFM plots and the Hainich plot inventory served as the basis to calculate annual harvesting and mortality as well as ingrowth rates to higher DBH classes.DBH class width was set to 4 cm and the data were summarized accordingly.Ingrowth rates were defined as the proportion of trees per diameter class growing annually to the next diameter class and were calculated directly from the annual diameter increment divided by the DBH class width (Schütz, 2006).All tree individuals that were not assessed in a consecutive survey were counted as removal by thinning, and we assumed natural mortality to be zero under the high management intensity in these selection forests.
To analyse ingrowth and removal patterns, we used multiple logistic regression to model the probability that a tree grows to the next diameter class or is removed in a certain year.A binomial distribution with a logit link was used, with the general form of the model according to James et al. (2017): where p i is the response probability for each observed tree i in DBH class d to be harvested or growing to the next DBH-class, and n is the degree of the polynomial function.Estimation of the regression coefficients (β 0 −β n )was performed using an iteratively reweighted least squares (IRLS) algorithm and maximum likelihood estimation (MLE) (Venables and Ripley, 2002).We fitted ingrowth and harvesting models with different degrees of the polynomial function and selected the models with the lowest value of the Akaike Information Criterion (AIC).The selected models were then used to predict the respective ingrowth and harvesting rates as a basis for the modelling of stem number distribution (cf.eqs (2) and ( 3)).
Furthermore, we analysed the ingrowth rates of juvenile trees at different light availabilities and compared beech and sycamore using mixed linear effects models with a nested random intercept given by the study site and the sample tree.To assess pairwise differences, estimated marginal means were calculated and adjusted by the Bonferroni method (Lenth, 2021).To evaluate the differences in the proportion of the two species at different light availabilities, a binomial general linear mixed model was fitted using study site as a random intercept.The P-values from the computed pairwise comparisons of the estimated marginal Figure 2 Annual ingrowth rates of juvenile trees calculated from the average diameter increment of the five last years at different light availabilities given by the GLI.DBH classes were defined by a width of 4 cm (1 = 0-3.9cm, 2 = 4-7.9cm, 3 = 8-11.9cm, 4 = 12-15.9cm).Similar letters indicate non-significant differences between groups and different letters stand for significant differences between groups on a P < 0.05 level from post-hoc pairwise Bonferroni test.
means of the factors tree species and light availability classes were adjusted using the Tukey's method (Length, 2016).

Modelling of stem number distribution
In uneven-aged forests, trees in many different diameter classes are present on small areas, and it is assumed that demographic sustainability can be achieved at the stand scale (O' Hara and Gersonde, 2004;Schütz, 2006).A model of the demographic equilibrium on the basis of stem number distributions was suggested by François (1938) and further developed by Schütz (2006Schütz ( , 1975)).The basic assumption is that a demographic steady state is reached when trees migrating to a higher DBH class replace the trees that were harvested and migrated out of the considered class (Schütz, 2006): where n i is the stem number in diameter class i, p i is the outgrowth rate and e i is the removal rate consisting of harvesting and mortality.Based on eq. ( 2), it is possible to model stem number distributions step by step forward from the lowest to the highest DBH class, or backwards from the highest to the lowest DBH class, as For DBH < 8 cm, tree-ring based growth rates were used (Figure 2) to model a case for trees growing at light availabilities below 10 per cent GLI (lower boundary) and between 10 and 20 per cent GLI (upper boundary).To model the stem number distributions for DBH > 8 cm, we used the lower and upper boundaries of the confidence intervals of the prediction of harvesting and growth rate (Figure 3) as input.Annual removal rates below the calipering  Sustainable regeneration in uneven-aged mixed deciduous forests limit were estimated to be within the range of 6-10 per cent (DBH 4-8 cm) and 16-25 per cent (DBH < 4 cm), based on mortality data from Bachofen (2009).

Determination of the demographic equilibrium
Starting from any arbitrary number of stems theoretically growing into the lowest diameter class of, say, 8-12 cm, different stem number distributions were modelled based on equation (2).Comparing the modelled relationship between stand BA and stem numbers in the lowest DBH class with observed numbers indicates the range of the target BA to maintain for demographic sustainability.
As a basis for the determination of the equilibrium, we exclusively used data from our field-campaign and Thuringia inventory plots that featured a stem number distribution with a positive skewness, so as to exclude even-aged stands.The plots were grouped in terms of their BA using a class width of 2 m 2 .Stem numbers were then averaged per BA class while stem number BA classes with <10 observations were not considered due to the high variability.
Harvesting rates were assumed to remove the total periodic increment required to maintain equilibrium conditions.Taking the harvesting pattern in equilibrium-like forests of Thuringia as a baseline, the absolute harvesting rates were multiplied by the proportion of the difference in increment between Thuringia and the respective EFM plot.
All data analysis and statistical work was conducted using R version 3.6.2(R Core Team, 2019).

Ingrowth rates and harvesting patterns
Annual ingrowth rates in the smallest diameter classes were highly variable with a positive trend along the light availability gradient.This trend differed between sycamore and beech.Below a light availability of 40 per cent, ingrowth rates of juvenile beech <8 cm DBH were almost constant (Figure 2).In contrast, sycamore of DBH > 4 cm showed lower annual ingrowth rates at light availability <10 per cent compared to beech and followed a clear trend towards higher values under better light conditions (Figure 2).
Patterns of annual ingrowth rates of trees with DBH > 8 cm were different at the EFM sites and the selection forest in Thuringia, where diameter increment was smaller (Figure 3).There was an increase of the ingrowth rates up to the diameter classes of 26-30 cm followed by a small decrease in the mid-sized trees at the Deinikon, Adlisberg and Hainich sites, where increasing rates were found for the larger size classes, culminating at a DBH of 80-86 cm.In Basadingen, a bellshaped curve peaking at 54 cm DBH represented ingrowth rates best, while the data for Muttenz were matched best by a monotonically increasing model (Figure 3, Table 1).
Harvesting patterns at the selection forest in Hainich were best represented by a U-shaped function (Figure 3, Table 1).At the Deinikon site, the harvesting patterns resembled those at Hainich but were not as high in the smallest classes.In contrast, at Adlisberg, Muttenz and Basadingen, harvesting rates followed different and more site-specific patterns, which were characterized by a steady increase (Muttenz), a steep increase from DBH 8 to DBH 24 cm followed by a decrease towards the large timber (Basadingen) and a small increase towards the smallsized timber at Adlisberg followed by a decrease in the mid-sized timber and an increase in the large timber.

Stem number distributions
Modelled equilibrium stem number distributions followed a typical shape characterized by a sharp decrease of stem numbers in the small diameter classes that flattened out in the midsized diameter classes and increased again for the largest dimensions (Figure 5).This pattern was more pronounced at sites with decreasing or constant growth rates in the mid-sized classes (Deinikon, Adlisberg and Hainich).The stem numbers of ingrowth in the DBH class 8-11.9 cm that are required to sustain this structure were between 95 and 174 per hectare, which in most study sites was congruent with current measured forest properties (cf. Figure 5).
While the observed stem number distributions for DBH < 40 cm corresponded well to the sustainability model at Hainich, observed stem numbers for DBH > 40 cm were higher.The forest stands at Deinikon, Muttenz and Adlisberg featured an excess in the stem numbers of the large timber (DBH > 50 cm), reflecting the ongoing transformation from former even-aged to uneven-aged stands.Modelled stem number distributions were approximated well by the EFM data from Basadingen (cf. Figure 5).
In the lowest diameter class (DBH < 4 cm), the stem number necessary to sustain a demographic equilibrium was between 500 and 3000 per hectare.At light availabilities below 10 per cent GLI, growth rates of trees with DBH < 8 cm were small (Figure 2), and therefore stem numbers at the respective DBH class needed to be larger to sustain a demographic equilibrium than at higher light availability, as reflected by the stronger decrease of the upper bound of the equilibrium range at DBH < 8 cm (Figure 5).At light availabilities between 10 and 20 per cent GLI, stem numbers of trees with a DBH < 8 cm sustaining the equilibrium were between 500 and 1000 ha −1 .

Table 1
Regression coefficients of the binomial logistic models with standard errors and P-values for the log-odds ratios to predict ingrowth and harvesting rates.The amount of deviance that the models account for is given as D 2 which is equivalent to the R 2 in linear models needed to warrant demographic sustainability, i.e. a stem number of 110 ha −1 in DBH class 8-11.9 cm (Table 2).In the smallest canopy gaps, sycamore exhibited smaller diameter increment than beech.In DBH class 4-7.9 cm, sycamore had higher diameter increment than beech in gaps >395 m 2 (Table 2).At low light availability (GLI < 10 per cent), there was a trend towards lower proportions of sycamore and higher proportions of beech (Figure 6) in the thicket class (1-8 cm DBH).At light availabilities between 10 and 29.9 per cent GLI, the median of the proportions from the total number of thicket trees was in a similar range, i.e. between 40 and 50 per cent for both species (cf. Figure 6).

Ingrowth rates
Patterns of ingrowth rates reveal important stand growth characteristics because they reflect management history, stand structure and species composition.In deciduous vs coniferous forests and in monocultures vs mixtures, stand development and dynamics differ (Pretzsch et al., 2017).Especially in view of the emerging popularity of selection cutting it is crucial to analyze the respective ingrowth patterns (Pommerening and Murphy, 2004;Brang et al., 2014).For uneven-aged forests managed by regular single-tree selection cuttings dominated by P. abies and A. alba, annual ingrowth rates were reported to follow a unimodal-shaped function of DBH (Bachofen, 1999), similar to site Basadingen in Figure 3 with a culmination in the mid-timber and large timber classes (DBH > 54 cm).However, in three of the five uneven-aged mixed deciduous forests studied here, ingrowth rates are not unimodal, but feature a plateau between DBH 30 and 60 cm followed by a second maximum in the large size classes, thus resembling a bimodal function (Figure 3).The different patterns observed can be explained by species composition shaped by past even-aged management and are not only important for our understanding of growth dynamics in selection silviculture, but also as a foundation of dynamic vegetation modelling (Schelhaas et al., 2018).
The predicted ingrowth patterns were more reliable for the smaller size classes than the larger size classes due to the rare occurrence of larger trees on small sample plots although their growth variability was smaller compared to mid-sized size classes.These large size classes exhibited the largest diameter growth as the deciduous beech can expand its crown continuously (Forrester et al., 2017), and thus quickly occupy more canopy space (Emborg, 1998).From that perspective, the competitive environment created by trees that are laterally closing the canopy and thus shade smaller recruits (Forrester, 2014) in combination with crowding effects helps to explain the plateau or slight decrease of ingrowth rates in the mid-sized classes (Canham et al., 2004;Bartkowicz and Paluch, 2019).

Forestry
Figure 5 Observations from the EFM plots, the Hainich inventory and the circular plots assessed in this study along with modelled stem number distributions (green shading) in a range of possible demographic equilibria.The upper dashed line for the DBH classes with midpoints of 2 and 6 cm represents growth rates at GLI < 10 per cent, while the lower line represents GLI 10-19.9 per cent.
It is unclear, however, to what extent phenotypic plasticity causes the large variability in the observed ingrowth rates, and how this is linked to resource availability and ontogenetic limitations (Ishii et al., 2013).Although it would be interesting to derive species-specific ingrowth rates of canopy trees to complement those reported for juvenile trees in our study, the sample  size of sycamore in the inventory and long-term monitoring plots was not representative to allow for reliable modelling.Except at very low light availability (<10 per cent GLI), juvenile sycamore of DBH < 16 cm were observed to grow relatively quickly, but there are indications that they lose vigour in the pole stage (20-30 cm DBH) and small timber classes (30-40 cm DBH) (cf.Hein et al., 2009;Brüllhardt et al., 2020b).Further research should therefore focus on the development of diameter increment of sycamore in uneven-aged forests to better understand the admixing possibilities of this (and possibly other) less shade-tolerant tree species.

Harvesting patterns
Managing uneven-aged forests to maintain demographic sustainability requires the regular harvesting of the accumulated increment using selection cutting.Understanding how these harvesting patterns influence stem distribution patterns is important not only for maintaining demographic sustainability from a management perspective (O'Hara and Gersonde, 2004), but also for adequately modelling the dynamics of managed forest stands (Lafond et al., 2014).The five examples in this study allow us to better describe and replicate harvesting patterns in mixed deciduous uneven-aged forests.
As harvesting rates per DBH class depend on the respective forest structure, the observed removal rates in the selection forests of Thuringia were assumed to be representative ('state of the art') to maintain sustainable deciduous selection forests that are dominated by beech.These patterns are similar to those in coniferous selection forests (Bachofen, 1999).In comparison, removal rates in the EFM plots are much lower in the small DBH classes due to lower stem numbers in the thicket class, caused by high stand BA.Thus, forest structure in the EFM plots is not in an equilibrium yet and therefore still influenced by the even-aged or coppice-with-standards management history, reflecting the long transformation time until demographic equilibrium structures are achieved (cf.Schütz, 2001b).

Demographic equilibrium
The determination of demographic equilibria revealed important relationships between ingrowth rates and stand characteristics.Due to the site-specific growing conditions, the relationship between stem numbers of the ingrowth (DBH class 8-11.9 cm) and stand BA was different for each site.This leads to the important notion that demographic equilibria depend on site-specific stocking volumes or stand BA (Bachofen, 1999;Schütz, 2006).Knowledge of the optimal target BA is therefore of crucial importance to determine harvesting rates (Cancino and von Gadow, 2002).
The values of target BA from our study that are required to maintain a demographic equilibrium are in line with values reported in the literature.Schütz (2006) calculated an equilibrium BA for the Hainich selection forests of 22 m 2 ha −1 , while a slightly higher 24 m 2 ha −1 was suggested by Hessenmöller et al. (2018).In forests dominated by shade-tolerant coniferous species, equilibrium BA was reported to be 28-32 m 2 ha −1 (Bachofen, 1999).

Forestry
With less shade-tolerant species and on less fertile sites, the target basal are to maintain a balanced structure is smaller, e.g.16 m 2 ha −1 in mixed P. taeda-P.echinata stands (Guldin, 2006).
In this context, the demographic equilibria determined for the presented sites in the Swiss lowlands have to be interpreted with care due to the unbalanced structure and the presence of large timber P. abies.Especially the higher estimations above 26 m 2 ha −1 are a result of the ongoing transformation from even-aged (Adlisberg) or coppice-with-standards (Basadingen, Deinikon) forests to uneven-aged stand structures.Although we tried to exclude such plots by filtering out stem number distributions with a negative skewness, the locally clustered occurrence of mature trees strongly influences stand BA and thus may blur the relationship between ingrowth stem number and BA.Therefore, the relationship between stem numbers of the ingrowth (DBH class of 8-11.9 cm) and stand BA varied broadly, and in some strata and particularly at low stand BA the number of observations was low.From a methodological point of view, a circular plot radius greater than 15 m (area ∼706 m 2 ) may have allowed for more reliable assessments of these highly variable properties.Another uncertainty lies in the lag time for trees to grow into the 8-11.9cm DBH class following thinning (Lundqvist, 2017), which may be of particular relevance for the transformation of stands from an even-aged to an uneven-aged structure.Nonetheless, the relationship we found is based on a more robust foundation compared with other studies (cf.Bachofen, 1999;Schütz, 2006).

Sustainable stem number distributions
Stem number distributions are important to evaluate the sustainability of selection forests, and they are pivotal as a solid and evidence-based foundation to guide forest management (O'Hara and Gersonde, 2004;Rubin et al., 2006).As tree age is of subordinate importance in uneven-aged stands, it is crucial to maintain appropriate tree numbers per diameter class to ensure continuous timber production and other ES.The stem numbers in the lowest diameter classes in particular are determined directly by forest structure via light availability (Canham et al., 2004;Ratcliffe et al., 2015).Therefore, it is essential to have a good understanding of the growth dynamics of ingrowing trees when aiming to develop sustainable stem number distributions.
To ensure the sustainability of the system, the observed ingrowth rates of sycamore in the lowest diameter class, with an average diameter increment of 0.13 cm at light availabilities below 10 per cent GLI (Table 2), suggest that ∼800-1700 trees ha −1 of a DBH < 4 cm would need to be present.As these sycamore individuals would be more than 30 years old when growing into the next diameter class, it is questionable whether they would keep their vigour or even survive such a long suppression period, given that their height growth vigour is ontogenetically limited (Hein et al., 2009;Brüllhardt et al., 2020b).Therefore, to maintain sycamore, higher GLI are likely to be required, which are equivalent to a gap size of minimally 400 m 2 .

Conclusion
The demographic structure of uneven-aged forests is mainly driven by the periodic harvesting of the volume increment, thus maintaining a stand BA that ensures continuous ingrowth.In mixed deciduous forests on beech site types, the equilibrium stand BA varies strongly, depending on site productivity and the proportion of coniferous trees.To maintain sustainable ingrowth of shade-tolerant species in beech forests, a target BA of 21-25 m 2 ha −1 should not be exceeded.
Within a homogenously spaced deciduous tree stocking of 21-25 m 2 ha −1 the annual DBH growth rate in the smallest DBH classes is often in the range of 0.1-0.2cm.In these conditions it is perfectly possible to sustain an equilibrium structure as long as: (1) stem numbers are large enough to replace the removals and outgrowing stems in the higher DBH classes and (2) timber quality is guaranteed.These conditions are met well by the growth strategy of shade-tolerant tree species like beech that is tolerating long suppression periods, but they are critical for the less shade-tolerant sycamore.
In terms of light availability, sycamore ingrowth may be promoted by a minimum gap size of 400 m 2 or a maximum canopy cover of approximately 80 per cent to ensure more than 10 per cent of above-canopy light.Therefore, to attain sustainable ingrowth rates and to increase the proportion of sycamore, group selection cuttings should be preferred over single-tree selection cuttings.

ForestryFigure 3
Figure 3Annual ingrowth (violet circles) and harvesting rates (orange triangles) per DBH class in the EFM plots and the beech selection forest Hainich, Thuringia, with predictions (ingrowth: solid violet line; harvesting: solid orange line) and confidence intervals (dashed lines) from the logistic models (Table1).

Figure 4
Figure 4 Determination of the demographic equilibrium based on the site-specific relationship between stand BA and the stem number of ingrowth (DBH class of 8-11.9 cm; dashed line with standard errors).The solid line indicates all theoretical model equilibrium states calculated based on equation (2).The theoretical optimum for the demographic equilibrium is where the lines intersect.

Figure 6
Figure 6 Proportion of sycamore and beech trees in the thicket stage (DBH 1-8 cm) at different light availabilities.Significant differences (P < 0.05) are indicated by different letters based on the results of the general linear mixed model.

Table 2
Tree ring-based annual diameter increment (± standard deviation) in DBH classes 1 and 2 (<8 cm) and DBH class 3 (8-11.9cm) of beech and sycamore at different light availabilities based on the GLI, canopy cover and gap size (± standard error) with respective modelled stem numbers to sustain an ingrowth of 110 stems to DBH class 3.