Stand Structure Impacts on Forest Modelling

Abstract

Modelling is essential in forest management as it enables the prediction of productions and yields, and to develop and test alternative models of silviculture. The allometry of trees depends on a set of factors, which include species, stand structure, density and site. Several mathematical methods and techniques can be used to model the individual tree allometry. The variability of tree allometry results in a wide range of functions to predict diameter at breast height, total height and volume. The first functions were developed for pure even-aged stands from crown closure up to the end of the production cycle. However, those models originated biased predictions when used in mixed, uneven-aged, young or older stands and in different sites. Additionally, some modelling methods attain better performances than others. This review highlights the importance of species, stand structure and modelling methods and techniques in the accuracy and precision of the predictions of diameter at breast height, total height and volume.

1. Introduction

Forest ecosystems exist over a long timeframe. Their assessment, monitoring and management requires information both quantitative and qualitative [1,2,3]. Stand structure has been used to characterise forest stands, as it provides information regarding its dynamics in time and space [4,5]. It comprises regime (characterised by the type of regeneration: high forest with the regeneration of seed origin and coppice with vegetative regeneration), structure (depicted by the number of cohorts or age classes: even-aged for stands with one cohort and uneven-aged for stands with two or more cohorts) and composition (refereeing to the number of species: pure for stands with one species and mixed with two or more species) [1,6,7]. Forest tree species, density, and tree spatial arrangement (both horizontal and vertical) are reflected in stand structure and its variability. The aforementioned derives in a wide range of interactions between individual trees, which in turn originates a wide suite of single tree allometries [8]. As a consequence, modelling is dependent on stand structure. The tree allometry variability has to be accommodated in the models in order to improve accuracy and precision and to minimise the prediction errors and uncertainties. This has resulted in an extensive set of models.

The focus of this review is the evaluation of the impact of stand structure on modelling at tree level. This study is organised in 5 sections; factors affected by and affecting stand structure (Section 2), reasons for modelling (Section 3), diameter at breast height and basal area (Section 4), height (Section 5) and volume (Section 6).

2. Factors Driving to Stand Structure Dynamics

Stand structure is dynamic in space and time and is influenced (and also influences) water and nutrient availability and uptake, light, climate, species, tree spatial distribution and interactions among individual trees. The higher the water and nutrient availability (and uptake), the higher the growth of a tree, due to the increase of leaf area index [9,10]. It also determines the aerial/root ratio. This ratio increases with the increase of the availability of water and nutrients, as the tree does not need to invest in the root system to uptake water and nutrients [11,12]. Inversely, in sites prone to drought and/or poor quality sites, trees have to invest in the development of the root system to ensure the uptake of water and nutrients [13,14,15,16,17,18]. Yet, water deficits can be mitigated by water hydraulic redistribution or by their use in a more efficient way [14,19]. Frequently, mixed stands with sociable species tend to have a more efficient use of water and nutrients due to the different uptake patterns, thus promoting complementarity rather than competition [20]. In short, aerial/root ratio tends to increase with the increase of water and nutrients availability.

The social status of a tree (i.e., their position in the canopy) determines the amount and quality of light reaching the crown. While the trees in the upper canopy receive more direct sun light, those in the lower canopy layers receive more diffuse light [21], the former have, in general, higher the leaf area index and growth. This is due to the direct proportionality between absorbed photosynthetically active radiation (APAR) and crown area, while inverse proportionality is observed between absorbed photosynthetically active radiation and crown length [22].

The absorbed photosynthetically active radiation and, consequently, the growth, is frequently higher in mixed stands than in pure ones [22]. This is due to different tree species phenology, which results in a reduction of competition [22]; different tree species growth dynamics and physiology (e.g., tolerance to shade) that result in the stratification of foliage [23]; wider range of the distribution of the foliage in the canopy [24]; wider range of variability in tree allometric relations, such as height, shape and crown [22]; and higher leaf area index and absorbed photosynthetically active radiation as a consequence of the differentiation in the horizontal plane [25]. The concentration of the crown in one layer in the canopy, especially in pure stands, results in higher competition for space and absorbed photosynthetically active radiation, and the crown sway and branch abrasion can originate in the reduction of the crown dimensions [22]. Additionally, species’ traits originate different allometric relationships for the individual trees, for example shade tolerance. Shade intolerant species in the lower canopy layers tend to reduce their crown (both width and length) and eventually die due to the lack of direct sun light, because of the shade cast by the trees in the upper canopy layers. Inversely, shade-tolerant species reduce their growth and the length of the crown but increase their diameter [26,27].

The forest microclimate, which is affected by stand structure and macroclimate, can too influence tree allometry. This is the result of the amount of water reaching the soil and temperature, which affects the soil-water evaporation rate, thus affecting water uptake and growth patterns of the trees [28,29].

Density influences tree allometry: the higher the density and the higher the irregularity of the spatial arrangement of the trees, the higher the variability in tree’ dimensions. This is basically due to two types of interactions: complementarity and competition. Mixed stands with sociable species tend to have complementarity interactions due to the different species traits [30]. Inversely, in pure stands, especially monolayers, competition tends to be higher as individuals have the same traits. This results in growth reduction and also variation in tree allometry, such as the reduction of diameter growth and of crown dimensions [9,25,30]. Tree spatial and temporal arrangements, including tree species proportions, create specific complementarity and competition patterns, which influence tree allometry. Regular, wider spacing enables free growth, with trees expressing their genetic traits. Irregular, denser spacing results in the increase of competition and in the variability of tree dimensions. At irregular spacing, the crowns might tend to be less symmetrical, their length depending on the available light and their width on the aerial available space [31,32].

Several silvicultural practices influence tree allometry as well. Pruning reduces the crown length and to a smaller extent of crown width. During thinnings, tree selection plays a key role both at tree level, allocating more growing space to the most desirable individuals, and at stand level, with the optimisation of the growth and yield [1,4,33]. Thinning increases growing space of the remaining stand and allows the increase of stem and crown diameter and, consequently, the variation of the trees’ allometric patterns. The method of thinning influences the trees to be removed from the stands. While in some methods the trees with the smaller diameters are removed (thinning from below), others select to remove the larger diameters (compensation thinning and thinning from above), and in others still, the selection is made in all the diameter range (selective or Schädelin and free thinning). Thinning can influence spatial arrangement, with some tending to a regular horizontal spacing (mechanical thinning) while others promote both horizontal and vertical irregularity (free and variable density thinning). The increase of the intensity of thinning tends to increase growing space and thus growth. The variation of tree allometry after thinning is due to the increase of diameter at breast height (larger annual growth rings), the reduction of the hd ratio (due to larger increments of diameter in relation to height), the increase of crown diameter and length (due to the increase of growing space, in particular light), and increase or reduction of crown eccentricity and stem inclination (when the constellation of neighbours is regular) (for more details see [34]). Additionally, fertilisation, increasing growing space, increases volume growth [35].

In short, tree allometry is highly dependent on stand structure and silvicultural practices. Overall, it can be said that tree allometric relations are variable in time and space due to a large set of factors. The incorporation of the variability of tree allometry is of the utmost importance to improving model accuracy and precision.

3. Reasons for Modelling

Due to the longevity and variability of forest stands, modelling is needed to assess and predict their development and yields in space and time. Several reasons can be pointed out for modelling, namely the labour and costs of forest inventory and its periodicity (between 5 to 10 years), as well as the fact that in some areas it is difficult to implement forest inventory due to stand structure and/or topography, and some tree and stand variables cannot be measured (e.g., volume). Models enable the prediction of tree and/or stand dynamics in space and time, and the development and testing of different models of silviculture which are not feasible in practice due to time and costs.

Traditionally, for modelling, the inputs correspond to a timeframe between crown closure and harvest [2]. However, the evaluation of the stand structure and silvicultural practices are of relevance before crown closure, which is the reason why sets of models have been developed (e.g., [36,37]). Additionally, models for pure even-aged stands originate in biased predictions for uneven-aged pure and mixed stands, which originated a new set of models (e.g., [38,39]). Again, regime influences stand an individual tree development which triggered the development of models for high forest and coppice regime (e.g., [40,41]).

Models are abstractions or simplified representations of some aspects of reality [42,43]. They represent an attempt to describe and simplify certain assumptions, determining afterwards their logical relationships [42] and are composed of one or several equations that predict dendrometric variables [2], competition, growth and yield [2,3,44]. Forest ecosystem modelling can be performed at different spatial and temporal scales, from cells to continents and seconds to millennia. The models can be less or more complex. Its complexity is determined by its goals and by the existing knowledge of the system [45].

Trees and stands have large variability due to a suite of factors, such as species, density, stand structure and site (cf. [2]). This led to the development of a wide range of models derived from diverse datasets (forest inventory, remote sensing and ancillary data, such as topography, soils and climate), with various mathematical methods and techniques. Models have increased in complexity, and frequently accuracy and precision, in time with the better knowledge of the systems; with the survey of more variables in the forest inventories; and with improved computer hardware and software capacities to compute large datasets.

Modelling goes far beyond applying statistical techniques to datasets and developing equations [2]. Forest models can comprise one or a set of equations that model growth, regeneration, mortality and changes in stand structure. The models provide an efficient way to predict forest resources, yet one of their primary roles is the ability to explore management alternatives [43]. Forest management is implemented utilizing a set of silvicultural practices from stand installation to final harvest. The initial stand structure, whether from natural or artificial regeneration, and the silvicultural practices prescribed (methods, intensity and periodicity) and their timing have a strong effect on the stand structure dynamics and individual tree allometry. The diversity of models of silviculture derives in a wide range of stand structures. It is not feasible to have research trials to characterise and describe all possible management options. Models, especially if implemented in simulators, particularly when they have visualisation software, enable the testing of many models of silviculture and serve as a tool to help the practice of silviculture [2].

4. Diameter at Breast Height and Basal Area

Diameter is always measured in forest inventories. Both diameter at breast height and basal area (corresponding to the area of the circle with diameter equal to diameter at breast height) are two key variables in the evaluation of the growth of trees, stands and forests. Forest inventories enable the evaluation of diameter (and basal area) variation with successive surveys. Yet, the drawback is that it is not possible to predict future growth, whether in time or in space. Thus, diameter and basal area models were, and still are, developed to monitor and predict growth. Several examples will be explored regarding the influence of stand structure on modelling at tree level.

Several authors have studied the influence of composition (pure vs mixed) on diameter models. Ref. [46] discussed model performance for young mixed stands of Picea abies, Betula pendula and Betula pubescens, subject to four different precommercial thinnings in Norway. The models for pure stands did not predict accurately diameter growth in mixed stands. The same authors stressed the effect of plot size on modelling. In small plots the contiguous trees (outside the plot) affected the trees in the plot and, consequently, its dimensions, and bias may arise in the estimation of diameter and dominant height. Ref. [47] and Ref. [48] stated that the different tree allometry in mixed stands, due to the variability between species, and thus growth, is better captured when models are developed per tree species. Ref. [49] studied diameter growth in even-aged mixed Picea abies, Abies spp. and Fagus sylvatica and pure Fagus sylvatica stands in twelve long term thinning trials in Slovakia. The authors compared nonlinear ordinary least square regression and nonlinear mixed effect models. The latter outperformed the former because it was better suited for data with hierarchical structure, which resulted in more accurate predictions and, at the same time, maintained biological plausibility. Similar results were attained by Ref. [50]. The mixed models advantages were twofold: they enabled the detection of the increase in the increments in diameter after thinning and the accommodation of the differences in diameter growth due to the species and their proportions, stand development stage and site quality [49,50]. Ref. [51] analysed diameter increment cores of 1240 dominant trees of long time series from both pure and mixed stands of Fagus sylvatica in thirteen European countries in mountain forests using mixed models. The models were able to accommodate the differences in diameter growth in pure and mixed stands (with higher slope for the former than for the latter) and along an elevation gradient (growth decreased with elevation). Ref. [52] modelled the radial growth on pure and mixed stands of Picea crassifolia and Betula platyphylla in China. The linear mixed effects models were able to accurately estimate the radial growth and differentiated it according to competition and diversity. The increase in competition resulted in the reduction of the radial growth for both species, radial growth depended on the tree size and that of the neighbours, and diversity in species positively affected the radial growth of Picea crassifolia, although the competition by the neighbours influenced the extent of the positive effects.

Other authors studied the effect of structure (even-aged vs uneven-aged) on diameter models. Ref. [53] evaluated the effect of selective cutting on diameter increment with forest inventory data and multiple least square models in Picea abies stands in Norway. The authors referred that the models developed for even-aged stands resulted in an overestimation of about 20% when applied to the uneven-aged stands. The bias was the largest after the selective cut when the diameter growth rate was the lowest. The model developed for the uneven-aged stands outperformed that of the even-aged stands and the variables with the highest predictive ability were diameter at breast height, total height and site index. Ref. [38] modelled the diameter growth for Pinus ponderosa uneven-aged stands with three levels of basal area in USA with field inventory data and a generalized growth and yield model (GENGYM). The function developed was compared with the models for even-aged stands frequently used by forest managers. The authors found that, at stand level, predictions were within the range of the precision of the measurements, while error at the individual tree level was 44% when compared with the even-aged functions; the model developed for uneven-aged stands improved in performance when crown ratio was included as an independent variable, and the error in basal area estimation was higher in the densest stand, probably due to the higher proportion of supressed (small) trees. It is likely that inaccurate estimates of diameter growth resulted in bias in the dynamics of stand structure in time, a point of importance where the target is attaining (or preserving) complex stand structures.

Ref. [54] fitted diameter increment models for Picea abies, Pinus sylvestris, Betula spp. and other broadleaved species in pure, mixed even-aged and uneven-aged stands in Norway with national forest inventory data and mixed models. The models for conifers, in general, outperformed those of the broadleaved species, probably because the latter comprised a group of species with different growth patterns. The stand structure was included in the model as a dummy variable, being able to differentiate the diameter growth patterns in even-aged and uneven-aged stands. One drawback encountered was the bias of the models, which might have been related to multicollinearity, due to the number of independent variables (between 10 and 13). Ref. [55] studied the effect of structure on the average basal area increment for six species in both even-aged and uneven-aged stands in Finland, with mixed models. The variability of growth due to species, stand structure complexity and site was accommodated in the models. The authors were able to detect similar net periodic annual basal area increment (PAI) in both structures, while for the even-aged structure PAI increased with the increase of growing stock, in the uneven-aged structure the relationship was curvilinear, and the stand structure complexity in even-aged structures enhanced PAI, while in uneven-aged ones it decreased (the latter probably due to the increase of the number of the small trees with low growth rates). Ref. [56] modelled the basal area for a natural oak forest with inventory data and nonlinear seemingly unrelated regression. Site quality, stand age, density and canopy layers were the predictive variables of the model. The models used either one canopy layer or three canopy layers, and the latter outperformed the former. The better performance of the model that included the canopy stratification was due to the different growth patterns in each layer. This is in conformity with Ref. [57].

Ref. [58] evaluated the effect of stand development stage on diameter growth patterns of Picea abies in Finland with national forest inventory data and mixed models. The author reported that mean diameter of basal area weighted by diameter distribution (d^Gm) described the development stage better than age, especially for the shade-tolerant species in low light levels (in shade). This is attributed to the species traits: shade-tolerant species are able to regenerate and survive supressed in the lower stand layers, not losing their ability to increase their growth rates after release. The same author demonstrated that mean tree size rather than age determines the patterns of growth, and for a certain d^Gm the variation of the growth patterns is small.

The influence of the silvicultural system on diameter increment was reported by Ref. [59]. In stands of Picea abies and Pinus sylvestris in Sweden, the effect of the shelterwood system, with two densities and no felling, were studied with forest inventory and stem analysis using generalised linear models (GLM). The authors reported that the model was able to capture the variability in diameter increment for the three treatments. Similarly, Ref. [60] evaluated the effect of three forest systems (clearcut with standards, basal area retention and no harvest) on basal area growth, with linear mixed models. The model enabled the identification of the variability of growth. In the no harvest treatment, the trees of the upper canopy layer (dominant and codominant) were the ones with the highest growth, while in the lower layers’ tree mortality occurred. The clearcut with standards resulted in a high density of regeneration individuals with high growth rates. In the basal area retention treatment, regeneration and recruitment occurred at a slower rate than in the clearcut one. The growth differences were due to the available growing space, which were higher in the clearcutting system than in the basal area retention one.

Ref. [61] studied the diameter increment of adult trees of Abies alba with increment cores in two areas of different elevations in France with linear mixed models. The authors’ main results were: competition explained 30% of the variability of diameter growth; a correlation between climatic variables (e.g., temperature and precipitation) and growth both from the present and former years; drought in summer (present and past years) reduced diameter growth; yearly spring temperature tended to increase diameter growth; drought that occurred in the previous years had a positive effect on diameter growth; variability in diameter growth per site (temperature, precipitation, relative humidity and elevation) was influenced by either the species plasticity or genetic adaptation; and growth reaction to summer precipitation had high variability, due probably to the variability (heterogeneity) of the micro edaphic variation. Ref. [62] (and references therein) demonstrated that low temperatures were the limiting factor of growth when elevation was high, while in the Mediterranean climates the limiting factors were drought and high temperatures. The authors observed that Fagus sylvatica growth diminishes with the increase of elevation because it is very sensitive to spring frost, while Abies alba had optimal potential growth at middle elevations, probably due to its sensitivity to the water deficits in summer. Ref. [63] evaluated the effect of drought events on the diameter increment of Picea abies and Fagus sylvatica stands in duplets with linear mixed effects models. The two species have different strategies to water stress. The mixed models were able to identify the variability in diameter growth, both the reaction per species and of both species, either under a drought event or not. Ref. [64], using increment cores and generalised additive models, analysed the effect of climate on the diameter increment of Picea abies. The model was able to capture the influence of precipitation and temperature on diameter growth. Precipitation promoted growth was denoted by the positive linear trend of the predictor (especially from May to July), whereas temperature had a non-uniform behaviour.

Several regression methods have been used to model diameter growth. Ref. [37] studied the effect of spacing on the diameter growth of Pinus ponderosa, Pseudostuga menziesii and Sequoiodendron gigantum in the USA with data from field plots and linear regression. The authors observed that there was a positive correlation between spacing (and, thus, growing space) and diameter at breast height for all species, which was probably due to the low competition levels between individuals. Another spacing trial was used by Ref. [36] with Sequoiodendron gigantum in the USA with field data plots and generalized linear mixed-effects models. In this study density had an inverse effect on diameter at breast height (the lower the density the higher the diameter at breast height) and a direct effect on height (the higher the density the higher the height). Ref. [65] modelled diameter increments for ten Quercus spp. in the USA with data from forest inventory and increment cores and mixed models under a Bayesian parametrisation. The selected modelling method was able to incorporate the variability of growth in time per and among trees, including the effects of growing space caused by density, resources and climate oscillation. Ref. [66] evaluated diameter growth with the stem analysis of Fagus sylvatica trees, with ages between 80–160 years, in Slovakia applying nonlinear least squares (NLS) models and Bayesian method with Kumaraswamy probability distribution (K function). The authors stated that the latter function had better predictive ability and that the distributions of probability derived in an enhanced biological interpretation, which resulted in backwards predictions with greater precision. One advantage of the K function was the incorporation of the asymmetrical growth in the model, with the parameter of likely life span. The Bayesian parametrisation, when compared with the NLS, had lower root mean square error (%) and lower bias for 2/3 of the fitted models.

Apart from the examples given above many others could be included. The examples highlight the variability of the diameter and basal area growth patterns at tree level. They are strongly dependent on composition, structure, silvicultural systems, site and density. Additionally, different regression methods and techniques have been used and those which accommodate the variability of growth have better performances.

5. Height

Height growth pattern differs by species, site and stand structure [67]. Height is frequently predicted as function of diameter at breast height. A set of functions have been used [2] (pp. 114,115). Due to the aforementioned factors, a wide set of h–d relationships has been developed, mainly for two reasons. First, because diameter is easier to measure more accurately than height [68]. Second, because height is used to predict other variables, such as volume [2]. Height functions should have monotonic increment, inflection point and asymptotical value, which corresponds to sigmoid functions [69]. The different growth patterns of trees by species and by site, as well as by stand structure, derive in a large number from height–diameter functions (h–d) (e.g., [58,67,69,70,71,72]). However, most of the functions were developed for monospecific or pure even-aged stands, many of which are monolayers. Thus, their use in multi-layered stands can result in biased estimations of the tree total height in uneven-aged stands [70]. Several authors (e.g., [58]) have demonstrated that some stand variables can improve the h–d function prediction. Considering the tree height and its growth variability, the flexibility of the h–d functions can enable more accurate height predictions (e.g., [70,73]). Traditionally, h–d functions were fitted using nonlinear least squares regression methods (e.g., [68,74]). These models are species-, site- and stand structure-specific. More recently, the inclusion of other variables and other regression methods enabled the accommodation of the variability in tree allometry, maintaining model accuracy. Some examples are used to illustrate the influence of the species, stand structure, site and regression methods on h–d functions.

Ref. [68] developed h–d functions for 13 riparian species for Missouri, Illinois and Iowa, USA, with nonlinear regression, demonstrating that the equations give reasonable predictions. However, the authors did not give information about the stand structure, indicating that the equations should only be used for the species and the regions where they were fitted.

For even-aged stands, Ref. [75] developed h–d models for Pinus taeda, for a range of densities using nonlinear regression with one or two independent variables (diameter at breast height, normalised using the Reineke stand density index, SDI). The authors demonstrated that the functions with only diameter at breast height were not able to detect the differences in stem tapper due to density, while the functions with the two independent variables had better predictive abilities. Additionally, for pure even-aged stands, but using mixed models, Ref. [72] stated that for Tectona grandis stands in India, the variability caused by age in h–d with mixed models was low, which was explained by the reduction of competition due to thinning. The effect of thinning intensity, while demonstrating wide variation, was incorporated in the random components of the model. Ref. [76], for Pinus pinaster, concluded that the function with mean quadratic diameter and basal area had good performance at stand level and that the predictive ability increased if the trees in the extreme range of distribution were used for the prediction of the random effect. Inversely, Ref. [74], comparing two regression methods (nonlinear regression and generalised additive mixed models) for Pinus massoriana in thinned (from below) and unthinned stands, concluded that the model attained using nonlinear regression outperformed the one fitted with the mixed model. The former was able to accommodate the changes in diameter and height growth of thinning (with thinning from below there was an increase of diameter at breast height growth and a reduction of height growth, the largest the heavier the thinning intensity) due to the reduction of competition.

According to Ref. [73], in mixed tropical forests in China, linear and nonlinear regression models had high precision. Ref. [70], discussing Pinus sylvestris, Picea abies, Betula pubescens uneven-aged stands in Finland, reported the good performance of the linear mixed models. The authors underlined the small bias and the flexibility of the model for different stand structures. Additionally, the independent variables used in the model were able to capture the tree and stand variability as random parameters, namely the stem form (north coordinate and elevation), tree allometry (stocking and thinning) and stand (absolute density measures). Ref. [49] discussed stands of Picea abies, Fagus sylvatica and other conifer and broadleaved species in Slovakia, where the h–d models included tree, stand, competition indices and dummy variables for species as explanatory variables. The authors demonstrated that the accuracy improved when the dominant height, ratio of tree diameter at breast height and stand basal area, and stand basal area were used as covariates and when the height of one to four individuals were used to estimate the random effects. Ref. [47] also stressed the flexibility of the mixed models of even-aged pure and mixed stands of Pinus pinaster and Pinus sylvestris in Spain. The fitted model was able to accommodate composition, thinning (from below with moderate intensity), vertical structure and site quality, which were reflected in the tree allometry. The model developed outperformed those for the pure stands and the reason for the better performance was due to the inclusion of the random effects. Additionally, using mixed models, Ref. [39] compared the performance of the models using inventory data and stem analysis data for Fagus sylvatica in pure and mixed stands, in France. The authors highlighted the advantages of modelling with data of growth measurements from stem analysis (dynamic relationships) in contrast with field inventory with a single measurement (static relationships), because in the former the information of the dynamics of tree growth was more detailed. The mixed models were able to detect the variability of growth in time. When silvicultural practices were performed, the trees from all social classes increased growth shortly after release (earlier in diameter than in height) and reduced growth gradually with the progressive canopy closure, the supressed trees were able to live for long periods with small or null diameter growth but continued to grow in height, and during the periods of release the individual trees showed wide variability in their diameter and height growth patterns. The flexibility of the mixed models made it possible to accommodate both social status and thinning. Likewise, Ref. [77] stressed that tree social status and competition had a high contribution to h–d generalised linear mixed models, regarding 44 species in France. The authors stated that: competition contributed more than temperature and precipitation; the mean temperature from March to September was an influential explanatory variable for 25% of the species, while precipitation was for 16% of the species; and the effect of temperature was stronger than that of precipitation. Similarly, Ref. [78] reported that nonlinear generalised mixed effects models outperformed the nonlinear regression for mixed uneven-aged stands in Romania. Again, the mixed models were able to capture the individual tree allometry variability, which is characteristic of the uneven-aged stands. Additionally, Ref. [79] compared two approaches with mixed models, additional random effects and co-modelling, with data of repeated measurements from forest inventories in Austria, with the goal of accommodating the temporal variability. The precision with additional random effects approach was the highest, while the co-modelling had the lower bias, and the approaches outperformed existing models.

Ref. [80] used field plot data to model h–d relationships in selection forests (uneven-aged) of Fagus sylvatica in Germany with quantile regression. The model’s good performance could be explained by stand structure: balanced diameter distribution with Plenter equilibrium, attained using periodical selective cuttings where trees of all diameter classes were cut; increment was centred in the trees with large dimensions (diameter and height); and the period of regeneration was long and the recruited regeneration growth rates were low.

Regarding natural stands of shortleaf pines in USA, Ref. [81] used inventory field data with linear and nonlinear mixed models. The latter outperformed the former, due to the incorporation of the variability of the tree allometry that resulted from artificial (thinning) and natural (storms) disturbances. The mixed models had better predictive ability than those of the existing ones; and the inclusion of mean quadratic diameter and dominant height improved the performance of the nonlinear mixed h–d model. Ref. [82] modelled the height for Pinus ponderosa in seven states of the USA with nonlinear mixed effect models. The authors demonstrated that the fitted model was able to accommodate the variability in tree allometry of site: higher temperature and precipitation promoted height growth, while drought restrained it. Regarding competition, the increase of competition increased the hd ratio, and in terms of species mingling, the lower the mingling of species, the lower the hd ratio.

The generalised mixed effects models have also been used to model h–d in young (3–19 years) Betula pendula stands of natural regeneration in Poland [83]. From the fitted models, those that included the largest diameters of the existing range as random effects created an improvement in the predictive ability of the model.

The variability of stand structure and tree allometry is continuously bringing challenges to modelling height. This includes the selection of the independent variables with the highest predictive ability, low collinearity and regression methods that enable the accommodation of the differences and variability between a wide range of stand structures, sites and species.

6. Volume

Volume is frequently modelled as function of diameter at beast height and height. As a result, it is affected by both diameter at breast height and height measurement precision and variability in tree morphology and, thus, allometric relationships. The following examples illustrate the diversity of functions.

Volume estimation functions have always played a primary role in the determination of productions and productivity. Many studies have developed tree level volume functions [35,49,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105]. Additionally, volume equations have been compiled by several authors [41,106].

Ref. [107] developed several volume functions for Picea abies stands in Norway, with nonlinear regression. The functions were fitted for pure even-aged and for mixed uneven-aged stands. The authors demonstrated that stand structure influenced function performance, with models fitted for pure even-aged stands attaining rather large errors when used for mixed and uneven-aged stands. Ref. [94] modelled the volume for Picea abies in Sweden using destructive sampling and verified that soil influenced mean annual volume increment and that existing models underestimated volume because soil variables were not included in the model. Inversely, the models developed by Ref. [94], including site index and soil variables, resulted in estimations with high accuracy. Ref. [108], modelling the volume for five species in Canada using stem analysis data with nonlinear regression, demonstrated better performances with the functions that used diameter at breast height and total height when compared with those that used only the former variable.

Ref. [103] modelled volume at national level for 25 most important species in Italy, with destructive sampling, using linear regression with a weighted function. The authors stated that volume estimates were unbiased. However, the authors pinpointed two problems: individual tree allometry variability caused by the geographical gradients coupled with the low number of sampled trees for some species might explain the error level, and, although weighted regression outcomes the residuals heteroscedasticity, error distribution did not meet the normality criterion; thus, in general, the intervals of confidence are not valid.

Regarding Pinus sylvestris and Picea sitchesis, using destructive sampling in the United Kingdom, Ref. [91] modelled volume through several statistical methods (linear regression, nonlinear regression and mixed models). In general, the nonlinear mixed effect models’ performance was better than that of the linear and nonlinear regression models. However, the former had a higher residual variance when compared with the latter, which might compromise its predictions.

Ref. [98] studied volume for two species, Pinus taeda and Eucaliptus spp., using data from the USA and Brazil, respectively. The authors incorporated the Hossfeldt diameter, Pressler height, the Kozak-type taper function and auxiliary variables in their developed formula. The developed volume equation was able to accommodate the variability in tree allometry and growth due to genetic improvement, site and silvicultural practices.

Ref. [101] modelled the volume for Pinus nigra, Fagus sylvatica and Quercus pubescens with data from the national forest inventory of Greece, using three functions: Hohendl–Krenn (independent variable diameter at breast height), power (independent variables diameter at breast height and height) and generalised least squares (GLS). The authors denoted that GLS had the best performance when the residuals’ variance was not constant; the diameter at breast height power equation and the combination of the diameter at breast height and height variance enabled the reduction of the residuals’ heteroscedasticity, and the weighted regression residuals met the normality distribution criterion. The authors stated that Hohendl–Krenn models were better suited to use at a forest operational level becausemost of the variance is explained by the models and diameter at breast height is more easily measured than height.

Ref. [35] modelled the volume for Pinus taeda in the USA in a fertilization trial (during the middle of the rotation) in stands with different densities and site qualities, using the maximum likelihood function. The advantage of the volume equation developed was that it accommodated the variability in tree allometry due to spacing and fertilisation, and the predictions biological behaviour were realistic. However, the random variation was only partially explained, and it could be overcome using the leaf area index.

Ref. [36] demonstrated that tree development and yield was dependent on the initial spacing, which was reflected in volume. Though volume is similar regardless of spacing, in narrow spacings, the volume per tree was lower than in wide spacings. Similar conclusions were obtained by Ref. [37].

Ref. [109] denoted that volume models in Norway for individual trees up to the recommended length of the rotation were accurate. The author demonstrated that the increment in volume for individuals older than the rotation age was equal or higher for stocked stands than that during rotation length, while it was inferior for understocked stands. The aforementioned originated the increase of the uncertainties of predictions, mainly due to the different behaviour of the mean annual increment (MAI) and current annual increment (CAI) beyond the rotation length (MAI and CAI curves converged in the models during the rotation length and CAI crossed the MAI curve beyond rotation length).

7. Conclusions

The longevity of trees and forest stands, the costs, labour and periodicity of forest inventories, along with the need to predict productions and yields and to develop and test alternative models of silviculture, stimulated the development of mathematical functions for diameter at breast height, height and volume prediction.

Many functions can be found for diameter at breast height, height and volume. They reflect the variability in tree allometry, which is dependent on the species, stand structure and site. The examples presented in this review highlight several approaches used, as well as their applicability to different species, stand structures and sites (

Table 1. Methods, advantages, disadvantages and performance of models to estimate diameter, height and volume.

Model	Advantages	Disadvantages	Performance
Linear and nonlinear regression	Easy to use	Species-, stand structure- and site-specific	Good performance, in general, for species, stand structures and sites for which they were fitted
Mixed models	Accommodates variability in tree allometry Flexible models	Large datasets needed Collinearity between independent variables	Good performance, especially for mixed- and/or uneven-aged stands Bias due to collinearity Better performance, in some cases, when random effects are included in the model
Generalized growth and yield models	Incorporated in a simulator Easy to use	Lower performance in complex structures Less flexibility when competition is expressed with distance independent variables	Slightly better performance when competition is included as distance-dependent variables Considerable better performance when crown ratio included as independent variable
Nonlinear, seemingly unrelated regression	Accommodates variability in multilayered stands Make compatible estimations for the total and per layer	Need to define thresholds for stratifying stand vertical layers	Higher performance than models per layer
K function	Enhances biological interpretation Enables backwards predictions	Requires long term datasets Need to formulate probability distributions	Better performance than nonlinear regression
Generalised least squares	Accommodates non-constant variance of residuals	Mode difficult to work at operational level	Better performance than nonlinear regression
Maximum likelihood function	Accommodates variability in tree allometry Flexible	Complex to fit	Good performance and biological plausibility

). Additionally, there is the need to evaluate the performance of existing models and, often, to develop new ones that are able to better accommodate diversity in tree allometry. This is achieved with the identification of the independent variables and regression methods and techniques that capture variability, resulting in more accurate and precise predictions.

It is recommended that future forest modelling should include a detailed characterisation of species, stand structure and site; the analysis of the accuracy, precision and errors of the models; and indications for the use of the developed functions at the operational level. Additionally, there is the need to have long term trials to evaluate the dynamics of stand structure in order to capture the variability of tree allometry. Moreover, detailed data on site and climate would enable the inclusion of independent variables that may improve the performance of the models.