Introduction

Being able to quantify tree growth spatially and over time is central to the management of forests worldwide. Tree growth is largely controlled by the prevailing stand abiotic and biotic conditions (Goudriaan 1977; Oliver and Larson 1996; Bonan 2008; Byun et al. 2013), including (1) available solar radiation, atmospheric and soil temperatures, wind velocities, soil water and nutrient content (Spurr and Barnes 1980; Botkin 1993; Oliver and Larson 1996; Kimmins 1997; Bourque et al. 2000) and (2) factors affecting inner-stand structure, competition and growing potential of individual trees (Weiskittel et al. 2011), including vertical stand composition, tree spacing, basal area (BA; Kahnamoie 2003; Coomes and Allen 2007; Watt and Kirschbaum 2011; Weiskittel et al. 2011; Huang et al. 2013; Mohammadi et al. 2018) and tree size.

Changes in tree growth allocation to height (H) and stem diameter (D) are a complex physiological, genetically programmed process (Becker et al. 2000; Jiang et al. 2016) that reflects the ability of trees to adapt to variable environmental conditions (Henry and Aarssen 1999; Koch et al. 2004; Poorter et al. 2006; King et al. 2009; Jiang et al. 2016). Trees have been documented to allocate their annual stem growth to H growth whenever resources are abundant and competition for sunlight places a greater significance on H than on D growth (Koch et al. 2004; Falster and Westoby 2005; Watt and Kirschbaum 2011; Harja et al. 2012; Trouvé et al. 2015). As trees continue to grow to maturity, annual H growth decreases in consecutive years mainly as a response to increases in hydraulic resistance and lowering of transpiration and photosynthetic capacity in taller trees (Oliver and Larson 1996; Ryan and Yoder 1997; Hubbard et al. 1999; Becker et al. 2000; Koch et al. 2004; Bonan 2008). Static H–D ratios in heterogeneous landscapes have also been known to vary with prevailing wind conditions, commonly decreasing with increasing wind speeds (Thomas et al. 2015). Increased swaying in tall trees triggers a predominance in improved radial growth at the expense of H growth in order to preserve the biomechanical stability of the stem (Meng et al. 2006).

In general, characterising stand conditions over large sections of landscapes is extremely difficult with field measurements alone, as the measurements may be taken far apart from each other in both space and time. Biotic variables suggestive of forest state, e.g. stand BA and density and individual tree H and D, are taken sporadically across the landscape, as most existing operational forest-monitoring networks have been created to answer forestry-related questions (for forest-management reasons) for regions spanning hundreds of thousands to millions of hectares (ha), as is common in North America, Russia and other parts of the world. Direct measurements of abiotic variables in situ are rarely included in such data captures. For this reason, capturing variability in forest landscapes at moderate spatial resolutions presents a major challenge. For this reason, combining forest plot data with computer-generated results of abiotic variables by means of evolutionary programming algorithms (machine learning) provides opportunities to advance our assessment, predictive capabilities and understanding of forest growth in heterogeneous landscapes at spatial resolutions relevant to on-the-ground operations common to forest and land management interests (< 100 m).

In recent years, there have been many attempts to describe abiotic attributes of landscapes by numerical methods (e.g. finite-difference modelling, artificial neural networks and spatial interpolation) and relate their values at point locations to tree growth measured in the field. Examples of analyses relating tree-growth, plot-level or photo-interpreted forest-cover descriptions to modelled abiotic quantities include those of Austin et al. (1996), Lebourgeois et al. (2005), Hassan and Bourque (2009), Byun et al. (2013), Ashraf et al. (2013), Baah-Acheamfour et al. (2013), Detto et al. (2013) and Bourque and Bayat (2015). Level of agreement between modelled and observed data generally varies with spatial resolution of the digital elevation model (DEM; Ashraf et al. 2012) and forest-cover data used in the analysis.

The principal objective of this study was to examine the abiotic and biotic controls on tree H- and stem D-growth and related patterns of H–D allometry in a heterogeneous, beech-dominated forest. Spatial patterns in beech development over a 9-year growing period, from 2003 to 2012, were related to (1) point extractions from computer-generated surfaces of growing-season mean potential solar radiation, seasonal mean air temperature, topographic wetness index, in representing variation in soil water and nutrient content and estimates of wind velocity generated with a computational fluid dynamics model (CFD) and (2) associated field measurements of tree H and D and plot BA (based on 2003 data) with the assistance of a system of simultaneous differential equations of tree growth and symbolic regression (Schmidt and Lipson 2009). The latter element is used in formulating species-specific functions of asymptotic tree H (i.e. Hmax, a site indicator of growth potential; Falster and Westoby 2005; Poorter et al. 2006; Banin et al. 2012) and cumulated leaf area. Tree leaf area is an important indicator of photosynthetic capacity in living trees (Thomas and Bazzaz 1999).

Methods

Study area and plot network

In northern Iran, the Hyrcanian (Caspian) forest covers approximately 50,000 km2, including the provinces of Golestan, Mazandaran and Gilan (Marvie-Mohadjer 2012; Fig. 1). Due to its humid, temperate climate and fertile soils, this region is renowned for its high forest productivity. Intensive human settlement in the lower elevations, as early as 1100 AD, has left large portions of the lowlands void of forest cover (Sefidi et al. 2011; Jaafari et al. 2014).

Fig. 1
figure 1

Forest plots essential to the current beech-growth study (a) close to Nowshahr, Iran. Also, indicated are the growing-season (April–October) wind directions recorded at the Nowshahr climate station (b); mean wind direction and standard deviation are indicated as a solid line at 333° from true North (0°) and error bar

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Beech forests in this region (Bayramzadeh et al. 2012) are mixed with Carpinus betulus, Alnus subcordata, Acer velutinum and several other tree species and shrubs. These forests are mostly broadleaved, but Taxus baccata and Cupressus spp. do appear on some specialised sites (Marvie-Mohadjer 2012). Close-to-nature silviculture (a management system based on small-scale interference and tree-group selection) is the harvesting method currently practiced in the region. It is similar in principle to forest-ecosystem management practiced in North America (Patry et al. 2013). This forest-management approach is best suited for establishing and maintaining mixed forests (Leibundgut 1983; Otto 1993a, b; Bayat et al. 2013) and permanent forest cover (Heyder 1986; Marvie-Mohadjer 2012).

The experimental forest (Kheyrud forest) is an 80-km2 unmanaged compartment of the greater Hyrcanian forest located about 7 km east of the port city of Nowshahr (36°39’N, 51°30′E; 7.5 m above mean sea level, AMSL; Fig. 1a). The northern lower boundary of the Gorazbon section, one of eight sections of the Kheyrud forest and location of the plot network (Fig. 1a), sits at roughly 1010 m AMSL; the section’s highest elevation lies at about 1380 m AMSL. The Kheyrud forest consists of 80 different tree species and 50 shrub species at various densities. Mean annual precipitation in the area is about 1303 mm based on data from 1985 to 2008, with October and August, respectively, being the wettest (235 mm) and driest month (42 mm) of the year. Mean annual air temperature is 16.2 °C, with February and August being the coldest (7.1 °C) and warmest months (25.1 °C). Mean annual pan evaporation is about 1031 mm, with highest average monthly evaporation occurring in August (155.4 mm) and lowest, in January (26.2 mm). Following the soil-taxonomic system of the United States Department of Agriculture, soils in the area are classified as highly productive udic alfisols. Organic matter accumulation on the forest floor is usually > 5 cm (Namiranian 2009).

The plot network in the Gorazbon section is designed on a rectangular grid (150 m × 200 m) and consists of 258 fixed circular plots of 0.1 ha each (Fig. 1a; Namiranian 2009), with 176 containing predominantly oriental beech and having GPS (global positioning system) coordinates for geo-referencing. Stem D measurements were taken with calipers in both 2003 and 2012 for all living stems of all tree species with a diameter > 7 cm. A collection of tree H were also made in 2003 on a sub-sample of beech trees (accounting for about 10.5% of 2246 sampled trees), with no follow-up measurements in 2012. Figure 2 describes the actual plot-level change in quadratic mean diameter at breast height of oriental beech (DBH and ΔDBH ± standard deviation) as a function of plot location (Fig. 2a) and DBH (or D; Fig. 2b).

Fig. 2
figure 2

Actual plot variation in quadratic mean diameter at breast height (DBH or D, in cm) and DBH (D)-increment (ΔDBH) for the study area (a). Background colours (shades of blue to cyan colours and browns) vary according to variation in a SAGA-based calculation of topographic wetness index (TWI, non-dimensional) for the plot-network-area in the Gorazbon section forest. Brown colours on the high-elevation plateau coincide with land depressions (vernal pools), where water has an opportunity to pool during the snowmelt season, i.e. TWI ~ 16. Low values (blue colour) coincide with drier soil conditions. A comparative graph of ΔDBH and DBH is provided in the inset (b). The circles around the individual points (quadratic mean diameter) are proportional to the standard deviation associated with individual-tree measurements. Isolated points, without circles, are associated with single tree measurements. (Color figure online)

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Tree-growth equation development

Here, we estimate individual-tree growth with respect to a coupled system of differential equations for tree H (m) and stem D (cm). The former equation is based on a differentiation of a static H-equation explicitly relating tree H to stem D. The D-increment equation predicts annual stem expansion anticipated under optimal-to-sub-optimal conditions as site-biophysical conditions (e.g. incident solar radiation, wetness index, seasonal air temperature and wind velocity) vary in space, given tree size (i.e. tree H and stem D) and leaf area. The two rate equations are linked through the asymptotic maximum tree H or Hmax.

The core assumption here is that as trees grow, the cost of maintaining a given volume of living tissue increases. As a result, the negative cost associated with growth causes tree growth to decelerate as trees get larger (Oliver and Larson 1996). Volume growth for unconstrained conditions, defined either as dV/dt or d[D2H]/dt, can be expressed as a differential equation of the continuous logistic function, such that

$$\, \frac{{{\text{d}}\left[ {D^{2} H} \right]}}{{{\text{d}}t}} = G \cdot {\text{La}}\left( {1 - \frac{DH}{{D_{\hbox{max} } H_{\hbox{max} } }}} \right)$$
(1)

(Botkin et al. 1972), where G is the intrinsic growth rate (cm3 of wood volume per m2 of leaf area), La is the cumulated leaf area (m2) and Hmax and Dmax are the maximum possible tree H (m) and stem D (cm) achievable under prevailing site conditions (Botkin et al. 1972).

Tree H is modelled as a function of stem D and a site-specific (pixel) calculation of Hmax. A three-parameter cumulative (asymptotic) Weibull function is used to model tree H at breast height (i.e. 1.3 m above the ground surface), such that

$$H = 1.3 + H_{\hbox{max} } \left[ {1 - \exp \left( { - b_{1} D^{{b_{2} }} } \right)} \right]$$
(2)

(Weiskittel et al. 2011; Ahmadi et al. 2013; Thomas et al. 2015; Ahmadi and Alavi 2013) where b1 and b2 are equation coefficients that determine the shape of the H-function as a function of D. The cumulative Weibull function is flexible and mathematically defined at its origin, i.e. D = 0, whenever H ≤ 1.3 m.

Given variable site conditions spatially in terms of incident solar radiation (SOL, kJ m−2), topographic wetness index (TWI, non-dimensional), seasonal mean air temperature (T, °C), wind velocity (WS, m s−1) and basal area (BA, m2 ha−1), we express the impact of site conditions on D-growth as a logistic function of site-biophysical variables to give a new site and plot-specific calculation of Hmax. Output from the logistic function, i.e. logistic[f(x)] → f(x) = 1.0/{1.0 + exp[−f(x)]}, where “x” can be any number of independent variables, varies from 0.0 to 1.0 as site conditions for oriental beech vary from optimal to sub-optimal. Maximum tree H in Eq. (2) (i.e. Hmax) is, thus, defined as

$$H_{\hbox{max} } = b_{3} \;{\text{logistic}}\left\{ {f_{1} \left[ {{\text{SOL}},{\text{TWI}},{\text{T}},{\text{WS}},{\text{BA}}} \right]} \right\},$$
(3)

where b3 is an equation parameter set during regression that defines the highest possible H oriental beech may reach under optimal growing conditions. Expansion of the left-hand side of Eq. (1) and differentiation of Eq. (2) with respect to time give

$$2DH\frac{{{\text{d}}D}}{{{\text{d}}t}} + D^{2} \frac{{{\text{d}}H}}{{{\text{d}}t}} = G \cdot {\text{La}}\left( {1 - \frac{DH}{{D_{\hbox{max} } H_{\hbox{max} } }}} \right),\quad {\text{and}}$$
(4)
$$\frac{{{\text{d}}H}}{{{\text{d}}t}} = H_{\hbox{max} } b_{1} b_{2} D^{{b_{2} - 1}} \exp \left( { - b_{1} D^{{b_{2} }} } \right) \cdot \frac{{{\text{d}}D}}{{{\text{d}}t}}.$$
(5)

Since long-term biophysical conditions across the landscape are, for now, considered static, Hmax as defined in Eq. (3) can be treated as a constant in the differentiation of Eq. (2). Substituting Eq. (5) into Eq. (4) and solving for dD/dt give

$$\frac{{{\text{d}}D}}{{{\text{d}}t}} = \frac{{G \cdot {\text{La}}\left( {1 - {{DH} \mathord{\left/ {\vphantom {{DH} {D_{\hbox{max} } H_{\hbox{max} } }}} \right. \kern-0pt} {D_{\hbox{max} } H_{\hbox{max} } }}} \right)}}{{H_{\hbox{max} } b_{1} b_{2} \cdot D^{{b_{2} + 1}} \exp \left( { - b_{1} D^{{b_{2} }} } \right) + 2DH}}.$$
(6)

Analogous to Hmax in Eq. (2), leaf area (La) in Eq. (6) is assumed to vary as a logistic function of site-biophysical variables, i.e.

$${\text{La}} = {\text{La}}_{\hbox{max} } \;{\text{logistic}}\left[ {f_{2} \left( {{\text{SOL}},{\text{TWI}},T,{\text{WS}},D,H - 1.3} \right)} \right],$$
(7)

where Lamax represents the upper limit in attainable leaf area, similar to b3 in Eq. (3). Combining the different equations, Eq. (6) can be rewritten as

$$\frac{{{\text{d}}D}}{{{\text{d}}t}} = \frac{{G \cdot {\text{La}}_{\hbox{max} } \left( {{{1 - D\left( {H - 1.3} \right)} \mathord{\left/ {\vphantom {{1 - D\left( {H - 1.3} \right)} {\left\langle {D_{\hbox{max} } b_{3} \;{\text{logistic}}\left\{ {f_{1} \left( X \right)} \right\}} \right\rangle }}} \right. \kern-0pt} {\left\langle {D_{\hbox{max} } b_{3} \;{\text{logistic}}\left\{ {f_{1} \left( X \right)} \right\}} \right\rangle }}} \right)}}{{\left\langle {100 \cdot b_{3} \;{\text{logistic}}\left\{ {f_{1} \left( X \right)} \right\}} \right\rangle b_{1} b_{2} D^{{b_{2} + 1}} \exp \left( { - b_{1} D^{{b_{2} }} } \right) + 2 \cdot 100 \cdot D\left( {H - 1.3} \right)}} \times {\text{logistic}}\left[ {f_{2} (Y)} \right],$$
(8)

where arrays “X” and “Y” involve variables SOL, TWI, T, WS and BA and SOL, TWI, T, WS, D, BA and H from breast height (above 1.3 m), respectively; “100” in the denominator converts metres into cm.

Abiotic surface development

Key to the development of the abiotic surfaces is the DEM of part of the terrain including the plot network (Fig. 1). DEM data are derived from the Shuttle Radar Topography Mission one arc-second (~ 30-m) Global DEM (https://lta.cr.usgs.gov/SRTM1Arc, last accessed on May, 2016), interpolated to 1-m resolution. Description of individual abiotic (environmental) surfaces (Fig. 3) and their derivation are as follows:

Fig. 3
figure 3

Growing-season distributions of mean abiotic conditions; different colours are associated with spatially varying intensity of the variable. Distributions include a growing-season mean incident solar radiation (kJ m−2 days−1), b near-surface air temperature (°C), c SAGA-based calculation of topographic wetness index (non-dimensional) and d wind velocity (m s−1). At the centre of each image is the network of forest plots, given for reference. Intensity of solar radiation varies from dark yellow (low intensity; ~ 4.3 kJ m−2 days−1) to lighter yellow (higher intensity; ~ 8.6 kJ m−2 days−1). Air temperature drops from mean sea level (~ 21.1 °C) to ~ 2200 m AMSL as a function of environmental lapse rate (i.e. 6.5 °C km−1). The pink in (c) represent wetter areas of the landscape, either as a result of the presence of wetlands, pools or low-ordered streams; the blues represent drier areas. The cyan and yellow in (d) represent areas of the landscape that are subject to low (~ 0.1 m s−1) and high wind velocities (~ 13 m s−1); areas with intermediate wind velocities are in pink and purple

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Solar radiation

Available solar radiation alters tree growth and tree distribution differently for different species. (Oliver and Larson 1996; Kimmins 1997). Sensitivity of oriental beech seedlings and saplings to various levels of incident solar radiation changes as plants mature (Tabari et al. 2005).

Incoming solar radiation in this study (Fig. 3a) is evaluated as a function of (1) DEM-based calculations of slope, aspect, view factor, horizon angle and terrain configuration factors, (2) sun-earth geometry and solar-illumination angles and (3) solar-flux calculations at the top of the atmosphere, based on calculations with the LanDSET (Landscape Distribution of Soil Moisture, Energy and Temperature) model of Bourque and Gullison (1998) and Bourque et al. (2000). Incoming sunlight at the top of the atmosphere is partitioned into its direct and diffused components as a function of incident angle, and each is handled differently as they pass through the atmosphere and interact with the underlying terrain, before being summed at the surface (Bourque and Gullison 1998). Many geographic information systems (commercial and open-sourced) have built-in subroutines to calculate cloud-free solar radiation spatially, including ERSI’s ArcGIS and System for Automated Geoscientific Analyses, or SAGA (Conrad et al. 2015).

Seasonal air temperature

Plant metabolism and growth increases with temperature (Nilsen and Orcutt 1996). For this reason, plant growth relates fairly well to indices of annual heat inputs. In this study, we use the growing-season mean air temperature as an index of growing-period heat input.

Seasonal distribution of air temperature (Fig. 3b) is based on decreasing the growing-season (April–October) mean air temperature at Nowshahr station (i.e. 21.2 °C; based on data collected from 1977 to 2005) as a function of DEM elevation and an assumed mean environmental lapse rate of 6.5 °C km−1 over land (Merriam 1992). The environmental lapse rate describes the long-term averaged drop in air temperature with increased elevation (Lutgens and Tarbuck 1998).

Topographic wetness index

Tree species differ in their soil water requirements and tolerances (Oliver and Larson 1996). As soil information and precipitation patterns for the study area were not available, we elected to represent soil water distribution as a function of TWI (Fig. 3c). Production of surfaces of TWI as proxies of soil water (and soil nutrient) distribution is justified, as precipitation and thus soil water are spatially redistributed by topography and accordingly surfaces of TWI can be developed from DEM-height data alone (Beven 1997; Maathuis 2006; Gruber and Peckham 2008). Topographic wetness index considers steady-state (long-term) conditions and spatially uniform soil–water infiltration and transmissivity. There are many methods available to calculate TWI. For specific information on these methods, refer to Gruber and Peckham (2008) and routines available in SAGA. In this study, we calculate the upslope contribution area in the calculation of TWI with the mass-flux method available in SAGA (Jaafari et al. 2018).

Wind velocity

An equally important abiotic variable affecting plant production is wind (Wadsworth 1959; Retuerto and Woodward 1992; Smith and Ennos 2003; Willoughby et al. 2009; Bang et al. 2010; Thomas et al. 2015). Wind velocities can have both positive and negative effects on plants, both from a physiological or biomechanical standpoint (Goudriaan 1977; Retuerto and Woodward 1992; Bang et al. 2010).

Wind is usually not considered in plant-growth studies, because of the difficulty in approximating its velocity and direction spatially. In this study, we use a CFD simulator to model wind flow over complex terrain (Fig. 3d), as specified by the DEM. The model solves the full 3-dimensional Navier–Stokes equations, which include the effects of atmospheric turbulence and thermal processes (Lopes 2003). Model calculations are based on a boundary-fitted coordinate system. Initial boundary conditions are specified by (1) the growing-season mean surface air temperature (the same as before) and wind velocity and direction (i.e. 1.7 m s−1 and 333° from true North; Fig. 1b) based on Nowshahr station climate records (1977–2005) and (2) an assumed wind velocity of 6 m s−1 at 500 m AMSL. In the calculations, atmospheric temperature stratification is assumed neutral (i.e. 9.86 °C km−1), a common state of the planetary boundary layer under windy and cloudy daytime conditions (Geiger 1965).

Surface verification

As it requires vast amounts of field data to verify computer-generated surfaces at enhanced spatial resolutions (i.e. at 1-m resolutions), verification at this level is usually not possible. Satellite data could be used to verify some of these images (in particular, incident solar radiation and air temperature), but not at the current spatiotemporal resolution without extensive image-data preparation and processing. Topographic wetness index and wind velocity are more challenging to quantify directly, in view of their undetectable characteristics. Satellite-derived soil water distribution could be used to verify TWI, but deriving soil water content from space is especially problematic under fully developed canopies as is present in the area. Albeit the lack of verification, we expect that the modelled abiotic surfaces provide reasonably realistic descriptions of the long-term physical conditions of the Gorazbon forest area.

Symbolic regression and application

Symbolic regression is founded on an evolutionary search (genetic programming) of algebraic equations that describe trends in training data. The search is directed with the minimisation of differences between target values and values calculated with the equations generated with the procedure (Schmidt and Lipson 2009). Different from traditional regression that determines coefficients of known equations, no specific mathematical expression is needed as a beginning point. Instead, primary expressions are formed by randomly combining base functions of input variables (linear or nonlinear) with numerical operators. Equations retained by the approach are those that replicate the target output data the best; undesirable solutions are simply discarded. The search for equations ends whenever the desired accuracy in training-data replication or machine learning has been reached, i.e. lowest Akaike information criterion (AIC).

Abiotic conditions (Fig. 3) at forest-plot locations were summarised separately as averages of values falling within individual 0.1-ha plots. For every plot containing beech (176, in all), plot records comprised of mean values of all abiotic variables, plot BA and individual-tree H and stem D of 217 trees and average annual stem D-increment (dD/dt or ΔD/Δt in its discretised form) over the nine-year growing period (2003–2012). Development of the coupled system of differential equations of tree growth (H and D) followed a two-step process (Fig. 4). The first step addressed the parameterisation of H (Eq. 2) and symbolic expansion of Hmax (i.e. Eq. 3) at the individual-tree-level given input of H and D (collected in 2003) and plot averages of modelled abiotic variables and stand BA (Table 1). The second step dealt with the development of the leaf-area component of Eq. (8) through symbolic expansion of Eq. (7) and specification of Lamax. Expansion of Eq. (7) was done within Eq. (8) and guided by plot-level estimates of D-increment, D (quadratic mean diameter at breast height), BA and H (modelled with an expansion of Eq. (2) obtained in step one, Fig. 4) and plot-means of the four abiotic variables. Parameters common to both differential equations (i.e. b1, b2, b3 and Hmax) were determined during the parameterisation and symbolic expansion of H and Hmax in step one of the procedure (Fig. 4 and Table 1). Numerical integration of the coupled system of differential equations (i.e. expansions of Eqs. (5) and (8)) was performed with a fourth-order Runge–Kutta integration of tree H and D with a 0.00125-year timestep, i.e. from 2003 over a 9-, 25- and 80-year time horizon.

Fig. 4
figure 4

Equation development process and information flow. Steps one and two of the process coincide with the development of the tree H-rate equation first, followed by the development of the D-rate equation. The H-rate equation is developed from individual-tree data and the D-rate equation from plot-level data. Information from step one, particularly values for b1, b2 and b3 and the symbolic expansion of Eq. (3) (Table 1), is passed on to step two in the derivation of the D-rate equation

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Table 1 Function definition for variable individual-tree maximum height (Hmax) in Eq. (2) [expansion of Eq. (3)] and plot-level mean cumulated leaf area (La) in Eq. (8) determined by means of nonlinear symbolic regression. Also specified are the equation coefficients of the coupled system of differential equations (e.g. G, Dmax, Lamax and bi, where i = 1,…,4) set a priori or during regression
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Results and discussion

Equation set development and performance statistics

Figure 5 provides a comparison of modelled and measured individual-tree H’s (2003) as a function of measured stem D’s. Overall explanation of variation in tree H with the fully parameterised version of Eq. (2) was 87.4%, with a root mean squared error (RMSE) and mean absolute error (MAE) of 3.60 and 2.84 m, respectively. Numerically based assessments of wind velocity and TWI across the study area provided the greatest overall impact on the calculations of H in oriental beech, with relative impact \(\left[ {{\text{i}} . {\text{e}} . { }\left( {{{\sum\nolimits_{i = 1}^{i = n} {\left| {{{\partial z} \mathord{\left/ {\vphantom {{\partial z} {\partial x}}} \right. \kern-0pt} {\partial x}}} \right|} } \mathord{\left/ {\vphantom {{\sum\limits_{i = 1}^{i = n} {\left| {{{\partial z} \mathord{\left/ {\vphantom {{\partial z} {\partial x}}} \right. \kern-0pt} {\partial x}}} \right|} } n}} \right. \kern-0pt} n}} \right) \cdot {{\sigma (x)} \mathord{\left/ {\vphantom {{\sigma (x)} {\sigma (z)}}} \right. \kern-0pt} {\sigma (z)}}} \right]\) of 38.3 and 37.3, respectively; for definition of terms, refer to the footnote of Table 2. Mean air temperature (T) and D provided weaker control on tree H (Table 2). Variables SOL and BA had no impact on the calculation of static H. In all instances, modelled abiotic variables related to H and Hmax in a nonlinear manner, giving both positive and negative responses to a monotonic increase in the variables.

Fig. 5
figure 5

Comparative graph of modelled and measured individual-tree H’s (in m) as a function of stem DBH (D, in cm). Asymptotic, maximum tree H (i.e. Hmax) for three sample calculations is indicated to the right of the graph. High values are assigned to sites of favourable growing conditions with respect to available soil water, represented here as a SAGA-based calculation of topographic wetness index (TWI), wind velocity (WS) and mean air temperature (T); lower values are assigned to lower-quality sites

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Table 2 Relative impact of independent variables (x’s) in the calculation of individual-tree height (H) for 2003 and mean annual diameter at breast height increment (i.e. dD/dt) in oriental beech over a 9-year growing period (i.e. 2003–2012). Independent variables are ordered from most to least important in the calculation of target variables (column 3). Height in trees without height measurements are estimated from Eq. (2) and Hmax, derived with the equation in Table 1
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The fully parameterised Eq. (8) was able to explain about 67.3% of target plot-level annual mean D-growth, with RMSE = 0.13 and MAE = 0.10 cm years−1. For the relative impact of the various predictor variables in the rate equation, consult Table 2. Equation (8)’s expansion by means of symbolic regression generated an expression of cumulated leaf area (Table 1; Fig. 4). The cumulated leaf area function (i.e. La) is central to the assessment of cumulated leaf area of beech trees as their stems vary in size and shape.

Impact of biotic variables on H growth

Maximum tree H, i.e. Hmax, is important here, because like site index in traditional tree-growth studies, it provides species-specific information regarding the photosynthetic capacity of trees (Thomas and Bazzaz 1999) and the asymptotic maximum H (i.e. growth potential) achievable under specific site conditions and site quality (Oliver and Larson 1996). Site index, in its conventional usage, is defined as the tree H one could anticipate at a specific reference age (e.g. 50 or 100 years) inferred from tree measurements (Weiskittel et al. 2011). Like its counterpart, Hmax integrates basic physiological information of tree growth as a function of site quality (Weiskittel et al. 2011). In the symbolic expansion of Hmax, it is clear that the maximum H of individual trees is exclusively controlled by the abiotic variables at individual tree sites (Table 2). Collectively, the abiotic variables were able to explain roughly 87% of the variation in static tree H (Tables 1, 2). Height in dominant trees tends to grow in a predictable fashion over a wide range of stand densities (Oliver and Larson 1996) and, as a result, variations in the current stand densities (i.e. ca. 200–700 stems per ha, averaging 298 ± 124 stems per ha−1) were altogether inadequate to evoke a noticeable influence on H-growth patterns under the current forest conditions.

Impact of abiotic variables on H–D change

In general, beech trees in areas of high wind velocities (> 10 m s−1) and low TWI (< 3, indicative of lower soil moisture), particularly along ridges, tend to promote the growth of shorter, wider-stemmed trees. The taller, slimmer beech trees and their associated growing environment tend to occur in parts of the landscape closest to sources of water, such as along streams and close to vernal pools, which regularly fill up with water during the snowmelt season. The vernal pools in Fig. 6b are illustrated by light brownish patches in the upper landscape with TWI ~ 16 and low prevailing wind velocities < 3 m s−1. Spring wetting of the well-drained soils and low-to-moderate wind velocities in the high-elevation parts of the landscape provide favourable growing conditions for oriental beech, causing H-growth in the trees concerned to increase relative to their radial growth (Fig. 7). This allometric relationship in vicinity to vernal pools is reflected in low ΔD’s (Fig. 2b), high Hmax’s (≥ 47.0 m; Fig. 6) and high ΔH:ΔD ratios (dH/dD in Fig. 7). Allometric patterns observed in this study are consistent with those reported in Fulton (1999), King et al. (2009), Watt and Kirschbaum (2011), Thomas et al. (2015), Trouvé et al. (2015), Navroodi et al. (2016) and Kalbi et al. (2018) for other tree species growing in different parts of the world.

Fig. 6
figure 6

Individual-tree H’s (red symbols of various sizes) overlain on a map of pixel-based (site) calculations of Hmax’s (Table 1) for the plot-network-area in the Gorazbon section forest (a); the different colours correspond to variations in Hmax (legend). The inset b depicts the same tree H data, but overlain a map of topographic wetness index (TWI). Light brown patches identified by the yellow arrows are land depressions (vernal pools) that collect water during the snowmelt season. (Color figure online)

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Fig. 7
figure 7

Ratio of ΔH:ΔD (red symbols of various sizes, legend) superimposed on maps of wind velocity (a) and topographic wetness index (TWI; b). In matching variation in wind velocity and TWI to colours on the maps, refer to the legend to the left of each map. (Color figure online)

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Cumulated leaf area

Figure 8a provides the modelled cumulated leaf area of oriental beech compared to measured leaf area of hardwood trees harvested in northern Cape Breton Island, Nova Scotia, Canada, and two oriental beech tree from the Kheyrud forest. In general, the shape of the cumulated leaf area curve, especially for smaller D trees (from 7 to 80 cm), is best displayed as a power function of D, or more specifically, leaf area (m2) = 0.099 × D1.90, with a coefficient of determination (r2) of 0.67, when all data were included (Fig. 8a). This power function is consistent with literature descriptions of cumulated leaf area in other forest systems, e.g. Vertessy et al. (1995). For larger D trees (≥ 80 cm), however, the cumulated leaf area becomes more unpredictable as a simple function of D alone, as the equation for cumulated leaf area in Table 1 illustrates. Environmental site variables become more important to cumulated leaf area as trees become larger.

Fig. 8
figure 8

Modelled and observed cumulated leaf area for oriental beech (m2) compared to observed leaf area for ten hardwood species from northern Cape Breton Island, Nova Scotia, Canada (a), and H:D ratio for oriental beech after 9 years of simulated growth (m cm−1) as a function of D (b). Scientific names of hardwood species not already mentioned in the body of the paper, include: Abies balsamea (balsam fir); Populus balsamifera (balsam poplar); Tsuga canadensis (hemlock); Quercus rubra (red oak); Betula alleghaniensis (yellow birch); Larix laricina (tamarack, larch); and Fraxinus Americana (white ash)

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Relationship between static H–D and ΔH-ΔD

Figure 8b displays the relationship between calculated H:D ratios at the end of the 9-year growing period (starting from 2003) as a function of D. In general, H:D ratios are largest when tree D is smallest (Fulton 1999). This basic property also holds true for ΔH:ΔD ratios (data not shown). In general, these trends are reinforced by Sumida’s (2015) commentary regarding Trouvé et al. (2015) research in Η-growth to D-growth interactions. These relationships (r2 = 0.77, for H:D ratios as a function of D) reflect the asymptotic property of tree measurements as tree size increases.

Figure 9 displays a strong linear relationship between H:D and ΔH:ΔD at the end of the 9-year growing period; r2 = 0.97 and associated p values < 0.05. This association suggests that the defining function between ΔH and ΔD is also asymptotic in nature, similar to the curvilinear relationship between H and D (Fig. 5) and consistent with the asymptotic relationship proposed by Trouvé et al.’s (2015) in their analysis of growth dynamics in sessile oak (Quercus petraea Liebl.; Sumida 2015). This relationship seems to maintain over the different time-integration periods (i.e. 9, 25 and 80 years) of this study (Fig. 9). Minor counterclockwise rotation of the nearly linear data clouds for the different time-integration scenarios may reflect inaccuracies caused by error propagation with long-term calculations. This error propagation may be an artefact of the less than perfect equations used to simulate the long-term interactions between static H–D and ΔH-ΔD. Despite the small inaccuracies, the coupled system of differential equations provides a direct linkage between static H–D and ΔH-ΔD relationships. The system also allows incorporation of site quality (abiotic variables, under the current forest conditions) as an influential feature in the computation of H- and D-growth dynamics in both space and time.

Fig. 9
figure 9

Tree H change relative to stem D change (i.e. ΔH:ΔD ratio) as a function of static H:D ratio based on tree-growth simulations over 9-, 25- and 80-year growing periods based on the integration of the coupled system of differential equations (Eqs. 5 and 8 after full expansion) with a fourth-order Runge–Kutta procedure and an 0.00125-year timestep

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Conclusions

The current paper presents a semi-empirical formulation of a coupled system of differential equations for the assessment of biophysical controls in the allocation of tree growth to tree H-growth and stem D-growth in an unmanaged oriental beech forest in northern Iran. The coupled system relates H and D-increment in oriental beech to plot values of computer-generated abiotic and field-based biotic variables, with an ability to explain about 87 and 67% of the variability in individual-tree and plot-level measurements and related variables. Analysis of equation sensitivity to the various input variables shows that changes in beech H and D are mostly controlled by abiotic factors, particularly wind and soil water content, through TWI. Large, mature trees are shown to grow more slowly, regardless of site variables. Observed patterns in growth allocation to tree H and radial growth relate to the asymptotic nature of both static H to D and Η- to D-growth relationships. Intra- and interspecific competition (modelled as a simple function of BA) exerts some level of control, especially with respect to D-increment, but its impact on H-growth is completely nonexistent relative to the impact of abiotic site variables. Reduced H-growth and increased D-growth (small H:D ratios) is observed more frequently in areas of the landscape with high wind velocities, particularly in windward and high-elevation parts of the study area (e.g. ridges), where drier soils predominate. Improved tree H-growth and reduced D-growth (yielding high H:D ratios) is more common in parts of the landscape with high soil water content, particularly in large, seasonally saturated depressions in the landscape (i.e. vernal pools) and where wind velocities are low. These results are consistent with the results in other studies of tree growing patterns in irregular terrain with variable site conditions. On the whole, changes in tree H relative to changes in stem D (i.e. ΔH:ΔD ratio) form a strong linear relationship with static tree H and stem D (H:D ratio; r2 > 0.97). The coupled system of differential equations described in this paper is sufficiently general to address the growing patterns of other tree species and forest-development patterns in other forested regions of the world by adapting expressions of potential tree H (by way of Hmax) and cumulated leaf area through symbolic regression. This type of work is strongly supported by the need to understand tree growth as it varies in both space and time, especially in view of the current forest-management needs and global climate change.