Power-law estimation of branch growth

We demonstrate the efficacy of power-law models in the analysis of tree branch growth. The models can be interpreted as allometric equations, which incorporate multiple driving variables in a single scaling relationship to predict the amount of growth within a branch. We first used model selection criteria to identify the variables that most influenced (1) the length of individual elongating annual shoots and (2) the total length of all elongating annual shoots in the individual branches of silver birch (Betula pendula Roth). We then applied the two resulting power-law equations as dynamic models to predict the trajectories of crown profile development and accumulation of branch biomass during tree growth, using total branch length as a proxy for biomass. In spite of the wide size range and geographical distribution of the study trees, the models successfully reproduced the dynamic characteristics of crown development and branch biomass accumulation. Applying the model to predict long-term growth of a single branch that was initiated at the crown top generated a realistic crown profile and produced a final basal branch size that was well within the range of field observations. The models also predicted a set of more subtle and non-trivial features of crown formation, including the increased rate of growth towards the tree apex, decrease in growth towards the lowest branches, the effect of branching order on the amount of elongation, and the higher vigour of thick branches when the effect of branch height was controlled. In contrast, a simple allometric model of the form Y=aX was incapable of capturing all the variability in growth of individual branches and of predicting the features of crown shape and branch size that are associated with the slowing-down of growth towards the crown base. We conclude that power-law models where the parameter a is refined to include spatial information on branch features shows good potential for identifying and incorporating actual crown construction processes in dynamic models that utilize the structural features of tree crowns.

We demonstrate the efficacy of power-law models in the analysis of tree branch growth. The models can be interpreted as allometric equations, which incorporate multiple driving variables in a single scaling relationship to predict the amount of growth within a branch. We first used model selection criteria to identify the variables that most influenced (1) the length of individual elongating annual shoots and (2) the total length of all elongating annual shoots in the individual branches of silver birch (Betula pendula Roth). We then applied the two resulting power-law equations as dynamic models to predict the trajectories of crown profile development and accumulation of branch biomass during tree growth, using total branch length as a proxy for biomass. In spite of the wide size range and geographical distribution of the study trees, the models successfully reproduced the dynamic characteristics of crown development and branch biomass accumulation. Applying the model to predict long-term growth of a single branch that was initiated at the crown top generated a realistic crown profile and produced a final basal branch size that was well within the range of field observations. The models also predicted a set of more subtle and non-trivial features of crown formation, including the increased rate of growth towards the tree apex, decrease in growth towards the lowest branches, the effect of branching order on the amount of elongation, and the higher vigour of thick branches when the effect of branch height was controlled. In contrast, a simple allometric model of the form Y = aX b was incapable of capturing all the variability in growth of individual branches and of predicting the features of crown shape and branch size that are associated with the slowing-down of growth towards the crown base. We conclude that power-law models where the parameter a is refined to include spatial information on branch features shows good potential for identifying and incorporating actual crown construction processes in dynamic models that utilize the structural features of tree crowns.

Introduction
Development of tree crown is an iterative process in which a tree constantly grows to adjust the three-dimensional (3D) structure in response to changes in the environment and the internal organization (Sorrensen-Cothern et al., 1993;Pretzsch, 2014). Although crown construction averaged over several tree species largely follows allometric laws and other biophysical rules (Farnsworth and Niklas, 1995;Enquist, 2002;Bejan et al., 2008), there remains ample variation within and between species and individuals (Lines et al., 2012;Bentley et al., 2013;Pretzsch, 2014).
The internal organization of a crown comprises multiple organs and tissues competing for limited resources. Allocation among the competing sites is affected by concurrent physical and ecophysiological factors that include hormonal control and correlative inhibition (Sprugel, 2002;Tworkoski et al., 2006), trade-offs among elongation, radial growth and reproduction (Obeso, 1997;Kramer et al., 2014), biomechanical requirements (Sone et al., 2006;Loehle, 2016), capacity T of transport tissue (Grönlund et al., 2016) and hydraulic limits (Koch et al., 2004).
Several studies of trees have reported consistencies between annual elongation and internal and external factors under strictly controlled growth conditions (Normand et al., 2009;Chen and Sumida, 2017), in tree saplings Takahashi et al., 2006, Collet et al., 2011, and in fixed crown positions (Osada et al., 2014). The generation of crown asymmetry is well documented, and the importance of hierarchical branching order for shoot growth is also well known (Jones and Harper, 1987;Young and Hubbell, 1991;Lintunen et al., 2011). However, the information available on within-crown patterns of branch growth is scattered, and spatially and ontogenetically limited, because typical studies cover growth at specific crown positions of only a handful of trees growing at a single site. In ecological interactions that shape the competitive success of individual trees, the functioning of plastic growth patterns within the crown thus remains largely unexplored (Ford, 2014). Even the process-based functionalstructural models of tree architecture frequently simplify representation of resource allocation to branches, using growth rules that overlook the identification of the actual processes that generate crown architecture (Lacointe, 2000;Mathieu et al., 2009;Ford, 2014).
Here, we focus on the analysis of branch growth throughout the entire crowns of silver birch (Betula pendula Roth) individuals ranging from the juvenile to the reproductive stage within a geographical gradient. Our aim is to identify the best set of explanatory variables for dynamic models that encompass the patterns of branch growth shared by the tree individuals regardless of the developmental stage or the effects of unknown site-specific factors. We considered both structural variables and variables that estimated the availability of photosynthetic radiation, as well as plausible proxies for physiological information. As an advancement to our previous analyses of static crown structure Lintunen et al., 2011), we demonstrate the use of power-law models as a method that serves both to detect the dominant factors influencing branch growth within the crown and as a method for constructing dynamic models to estimate branch growth and crown formation.

Power-law modelling
The power-law models can be considered as allometric scaling equations (Y = aX b ) in which the normalization constant (a) has been modified to include information on additional factors affecting the scaling relationship, a procedure that has been implicitly adopted in many existing model equations (e.g. Cole and Lorimer, 1994;Mäkinen, 2002;Kantola and Mäkelä, 2004), and explicitly considered in others (White and Gould, 1965;Kaitaniemi and Lintunen, 2008;Peters et al., 2018).
We analysed branch growth within the crown of silver birch by focusing on annual branch elongation rate described as 1) the annual total branch elongation (ΔL b ), calculated as the summed length of all elongating shoots within individual first-order branches, and 2) the annual length of the individual elongating shoots (ΔL s ). The models for ΔL b and ΔL s were constructed as power-law models, imitating the Ssystem approach (Voit, 2000), as: where X i are independent explanatory variables, α and g i are parameters, and j = b refers to the model of ΔL b and j = s to the model of ΔL s . Using branch diameter (d b ) as an example of an explanatory variable (X gi ), the Eq. (1) can be interpreted as an allometric equation with the form where g i is the scaling exponent and a is the normalization constant following from Eq. (1) as The construction of power-law models is not restricted to structural features, which are widely used for the static allometric analysis of tree structure by utilizing logarithmically transformed versions of powerlaw equations (Sileshi, 2014). Variables describing the contribution of physiological processes can be equally included in the list of independent explanatory variables (X i ) as long as their values can be considered constant at the time scale when the value of the dependent response variable (such as ΔL j here) is set (Voit and Sands, 1996). Yet, the potential of power-law models to operate as dynamical models of plant development by themselves has remained largely unexplored, with few examples covering the dynamic processes of tree growth (Voit and Sands, 1996;Renton et al., 2005a,b). In other contexts, the Ssystem approach with its power-law notation has been documented extensively with plentiful justification for process modelling (Savageau, 1979a,b;Voit, 2000;Voit et al., 2015).

Variables for models of branch elongation in silver birch
We considered a selected set of structural and physiological variables to potentially influence branch growth in silver birch ( Fig. 1, Table 1). We first used a full set of candidate variables to construct models according to Eq. (1), and then used model selection criteria to identify the most parsimonious set of variables to remain as independent variables X i . One year was used as the time step to calculate ΔL j and to estimate the values of X i . Branch-specific values were used for ΔL b , and values specific for an individual long shoot ( Fig. 1) were used for ΔL s as described in Table 1.
The annual levels of PAR and N area were included as potential X i variables, because both have been associated with crown growth habits  Table 1. h rt = h 1 /h t , where h 1 is the height of the branch base and h t is the total tree height. h rc = h 2 /h c , where h 2 is the distance between the branch base and the lowest living branch and h c is the crown length. l rb = l/l b , where l is the distance between the shoot base and the branch base along the line l b , and l b is the primary branch axis length measured as a straight line between the branch base and tip. d bh is diameter at breast height, d b is the primary branch base diameter and d s diameter at the bifurcation point of the side branch bearing a long shoot. In silver birch, the long shoots are responsible for elongation and production of new buds whereas the short shoots bear only one bud and typically elongate less than 5 mm per year. The short lines (|) crossing the branch indicate typical points of recording information during the measurement of branch structure. The downward facing side branches have been drawn without foliage.
in different studies (Umeki and Seino, 2003;Dong et al., 2015, Coble et al., 2017. PAR and PAR av (Table 1) indicate the potential to capture photosynthates, and N area indicates the availability and distribution of a key nutrient that also itself is related to the photosynthetic potential within the crown (Evans, 1989;Wyka et al., 2016). Although the values of PAR and PAR av correlate with N area , their relationship in silver birch is not straightforwardly linear, which leaves open the possibility of partially independent effects .
The availability of growing space and the risk of mechanical abrasion within the stand are also both known to influence growth allocation (Young and Hubbell, 1991;Hajek et al., 2015), and their potential effects were included in the models using l bn to indicate the proximity of neighbouring trees.
The variables d b , d s , and d bh , or variables closely related to them, are standard variables in models of plant development, and their influence on elongation growth can be linked to both structural allometry and transport capacity within a tree (Ford et al., 1990;Grote and Pretzsch, 2002;Savage et al., 2010). d bh also serves to indicate the age and ontogenetic stage of a tree.
The annual values of the variables h rt , h rc , and l rb , in turn, represented the variants of positional variables, which have turned out successful in characterizing many patterns of tree growth (Baldwin and Peterson, 1997;Grote and Pretzsch, 2002;Remphrey et al., 2002, Renton et al., 2006Normand et al., 2009). They can be considered as aggregate variables that potentially contain information on many types of processes. For example, they are, to some extent, linked with the distribution of PAR within the silver birch crown , but may also reflect various features of internal organization in terms of the relative transport capacity for nutritive, hormonal or other substances within the stem, crown or individual branch (Sachs, 2004). In the analysis of ΔL b , their average values also reflected structural constraints stemming from previous growth events.
We excluded branch mortality and other types of resource loss, due to the lack of substantial data, and likewise did not consider the numbers of buds produced within branches.

Data for model calibration
The data for the calibration of the branch growth models originated from eight study plots with two to six trees each (N = 32 trees in all), and were a subset of the data described in further detail in Table 1 from both Kaitaniemi and Lintunen (2010) and Lintunen and Kaitaniemi (2010). The sample trees had either Scots pine (Pinus sylvestris L.) or silver birch as the dominant neighbouring species. Already existing architectural models were used in the estimation of PAR and PAR av for these two species . The 4 -35-year-old study trees were in the phase of active growth and covered a wide size range ( Table 2). The study plots were located along a 400-km southwestnortheast transect between latitudes 60°N and 63°N and longitudes 21°E and 29°E in the boreal forest zone of Finland and represented Myrtillus type forest sites, characterized by mesic till soils and medium fertility.
Annual elongation through the growth of long shoots was measured to describe the rate of elongation in each study branch (ΔL s and ΔL b ). Elongation was measured at the end of the growing season from a sample of 1-12 sample branches systematically selected from the basal, middle and apical part of each tree crown. On average there were six sample branches per tree. ΔL s was measured as the distance between the long shoot base and the most apical bud of a long shoot, and ΔL b was calculated as the sum of all individual long shoot lengths within a branch (Fig. 1).
The ΔL s values for each long shoot position were obtained, along with the full 3D structure of the sample branches, by recording the 3D positions of all individual short and long shoots and branching points using a digitizer in the field (Fig. 1, Lintunen and Kaitaniemi, 2010). Dead side branches were left out from the measurements. The procedure provided also the 3D positions of the sample branches within the main stem. The branching orders (i) and the lengths of branch segments between the long shoot positions were also recorded, which yielded positional information for model development (Fig. 1). In addition to the detailed measurements of the sample branches, the length l b and branching angle for all the remaining branches of each tree was measured using the digitizer (Lintunen et al., 2011).
Within each sample branch, N area and relative PAR were estimated for one to two foliage sampling positions depending on branch size. Leaf samples for the determination of N area in the lowest crown parts mainly originated from the nonelongating short shoots, whereas the samples in the uppermost parts frequently included two fully expanded basal leaves of elongating long shoots. The leaf area was measured digitally from fresh samples, and the amount of N was analysed from dried samples with the Kjeldahl method. PAR was estimated for the leaf sampling positions , using the LIGNUM model as described in Lintunen et al. (2013). The instantaneous relative PAR with respect to the PAR of an open overcast sky was used as a shortcut estimate of the annual cumulative PAR (Gendron et al., 1998;Yoshimura and Yamashita, 2014).
In order to estimate PAR for the positions of leaf samples, the 3D crown structure with leaves of all trees on the plot was constructed using the models described in Lintunen et al. (2011). If the sampling area-based nitrogen content of foliage l bn cm tree-specific average distance of basal branches to neighbouring tree branches a Sampled within a cube with 20 cm side lengths around the shoot base. position was located inside the canopy, PAR was computed for a total of four random positions within a cube with 20 cm side lengths around the sampling point, and the average of the values was used in the analyses to even out the fine scale variation of light in the crown . Branch-specific average values of relative PAR (PAR av ) and N area (N area, av ) were used for the analysis of ΔL b , whereas the analysis of ΔL s included only those long shoots (on average nine long shoots per tree) for which measurements of N area and PAR were available within the same 20-cm cube where their basal point was located . Branch base diameter d b was measured at the base of each primary branch axis, and diameters (d) for the remaining branch parts, including d s , were estimated with a taper model (coefficient of determination R 2 = 0.96) based on unpublished measurements collected during the study by Lintunen (2013): where i is the branching order, such that for the primary branch i = 1, for side branches bifurcating from the primary branch i = 2, and so on.
In the estimation, we assumed that the minimum acceptable diameter was 0.15 cm. The minimum distance between the tips of the longest branches and the branches of each crown-bordering neighbour tree was measured in four 90-degree compass sectors to obtain tree-specific average distance of basal branches to neighbouring tree branches (l bn ). The average of l bn was used for all branches and all long shoots within branches. The measurements of l bn were restricted to neighbouring trees with stems within a 5-m radius from the target tree stem.

Model selection
We first parameterized the model in Eq. (1) for both ΔL b and ΔL s with all the candidate explanatory variables (Table 1) included, and then proceeded with backward elimination of the variables that did not contribute to the model fit. Occasionally, when elimination occurred between two almost equally important variables, the variables eliminated were added back at a later stage to ensure that the contribution of a particular variable was not dependent on other variables remaining in the model. Model selection and parameterization was conducted using the SAS procedure NLMIXED (SAS Institute Inc., Cary, NC, USA) with the Newton-Raphson method with line search as the optimization method. Normal distribution was set as the conditional distribution for the dependent variables ΔL b and ΔL s (Wolfinger, 1999). Random effects were not considered because the focus was on the selection of fixed explanatory variables and their parameterization. The Bayesian information criterion (BIC) was used as the criterion for model selection (Schwarz, 1978). The steps of model selection are shown in Appendix A.

Model validation and analysis
We utilized model-predicted trajectories of long-term branch elongation, and the resulting estimates of crown profile and branch biomass, to validate the behaviour of the final models against three sources of additional data. In the validation, we assumed that the total branch length (L t ) at any time step was suitable as a proxy for branch biomass. The details of calculating the crown profile and the trajectories of growth using the models of ΔL b and ΔL s are described in Appendix A.
The first source of data for validation was the model of proportional branch biomass growth by Tahvanainen and Forss (2008), which provided an independent estimate of branch biomass increment over time in different crown positions. We repeatedly applied the model of ΔL b to generate a similar trajectory of biomass increment as Tahvanainen and Forss (2008). The trajectory of growth was obtained by using the final model of ΔL b to predict the growth of a single branch initiated at the crown top.
The second source of data was the L t of the sample branches, which was also obtained during the digitization of the 3D branch structure (Fig. 1). The measured L t provided estimates of L t at different crown positions and that way reflected the prior trajectory of total branch elongation (ΔL b ) in the field. To validate the final model of ΔL b , the average field-measured L t was compared with L t predicted by the model at different crown positions during the crown development, similar to the estimation of the biomass growth trajectory.
The third source of data were the lengths (l b , Fig. 1) of the remaining branches in our study trees, i.e. those branches that were not used for model parameterization, to estimate the average crown profile of the study trees. To obtain a model-predicted crown profile for comparison, the model of ΔL s was repeatedly applied to predict the branch main axis length (L c ) at each annual time step and crown position. Because the model of ΔL s predicted L c , whereas l b in the data underestimated L c by ignoring curves within a branch (Fig. 1), the crown profile for comparison was generated from l b by estimating L c = 1.18l b (see Appendix A for more details). For simplicity, because the crown profile was obtained as a by-product of comparing the model predicted and observed branch lengths during the crown development, we assumed a constant 45°branching angle and a linear shape for all branches in the comparison. Description of a more accurate crown profile estimation for silver birch is available in Lintunen et al. (2011).
Finally, a more detailed analysis of model behaviour in the various crown parts and in trees with differing size was obtained graphically by plotting the outcomes of altering the values of X i within a representative range.

Alternative models
The performance of the final models of ΔL s and ΔL b was further assessed by comparing them with alternative models obtained from the literature. The fit of the alternative models was compared with the measured ΔL s and ΔL b using the same model diagnostics as with the power-law models. In addition to BIC, the fit was assessed by analysing the regression between the observed vs. the predicted values and testing the significance of slope b = 1 and intercept a = 0 (Piñeiro et al., 2008). The root-mean-squared error (RMSE) was also reported for the alternative models. Further, the fit was also assessed by using the alternative models to generate similar growth trajectories as described in Model validation and analysis.
First, a simple allometric model was considered as plausible alternative for ΔL b : Second, we used the empirical model underlying the statistics in Table 2 of Lintunen and Kaitaniemi (2010) to infer ΔL b from the statistical model explaining variation in growth vigour (GV). Lintunen and Kaitaniemi (2010) measured GV as totlong t /totlength t-1 , where totlong t is the measured total annual elongation at the end of year t and totlength t-1 is the total length (including all alive side branches) at the end of year t-1. Since GV in Lintunen and Kaitaniemi (2010) was based on tree level sums of lengths, including stem elongation, we assumed that the individual branches grew with the same GV as whole trees, resulting in where CI 12 is a competition index, p 1 = 0.03 and p 2 = 0.02 with Scots pine neighbours, and p 1 = 0 and p 2 = 0.03 with silver birch neighbours. The measured values of input variables  were used in the analysis of ΔL b whereas the generation of growth trajectories relied on the average GV observed in the P. Kaitaniemi, et al. Ecological Modelling 416 (2020) 108900 data. Third, the model predicting ΔL s and underlying the statistics in Lintunen and Kaitaniemi (2010) was also applied to the subset of trees and shoots present in our sample where CI 9 is a competition index, L c is equal to l b , p 1 = 0, p 2 = 0 and p 3 = 0 with silver birch neighbours, p 1 = 0.86, p 2 = -0.51 and p 3 = -0.06 with Scots pine neighbours, p 4 varies as a function of i as p 4(1) = 3.4, p 4(2) = 3.6, p 4(3) = 3.9 and p 4(4) = 3.3 with p 4(1) being the parameter for primary branch etc, and p 5 correspondingly varies as a function of i as p 5(1) = 0.18, p 5(2) = 0.04, p 5(3) = -0.05 and p 5(4) = -0.01. The generation of crown profile, by predicting the ΔL s of the primary branch, was based on the observed average values of CI 9 and treeage and the assumption of silver birch neighbours . The values of h rc and L c for the generation of crown profile were calculated as described in Appendix A. Eqs. (6) and (7) both describe statistical models with interactions between continuous variables, which requires the values predicted to be calculated as described, e.g. in Jaccard et al. (1990). We used the SAS procedure GENMOD to generate the predicted values as an output.

Performance with calibration data
The variables X i retained in the final model for total branch elongation ΔL b (Table 3a) included the height of branch base in relation to crown length (h rc ), branch base diameter (d b ), and branch-specific average values of both relative distance of long shoots from branch base along the main axis (l rb, av ) and side branch diameter at bifurcation points (d s, av ). The final model for individual long shoot elongation ΔL s (Table 3b) also included the height of branch base in relation to crown length (h rc ) and branch base diameter (d b ), together with the diameter at the bifurcation point of the side branch bearing the long shoot (d s ).

Performance with validation data
The model of ΔL b produced a growth trajectory for L t that was closely similar to both the trajectory of total branch length observed in the field (Fig. 4a), and the trajectory for branch biomass resulting from the model by Tahvanainen and Forss (2008) (Fig. 4b). Predictions based on GV in eq. (6) failed to follow the observed values. The simple allometric model of eq. (5) (with parameters α = 38.4, β = 0.26) succeeded well in predicting L t , although it did not track the curvilinear trajectories predicted by both the model of ΔL b and Tahvanainen and Forss (2008) (Fig. 4).
The model of ΔL s applied to predict the growth trajectory of the branch main axis resulted in an approximate crown profile that followed the observed branch lengths, whereas the model in eq. (7) overestimated branch lengths (Fig. 5).

Model predictions
The parameter values in the final models (Table 3) suggest that h rc and d b both have a strong effect on ΔL b (Table 3a). Further, ΔL b was highest in branches with low branch-specific average side branch diameter at the bifurcation points within a branch (d s,av ) and low branchspecific average distance of shoots from the branch base along the main chord (l rb,av ). ΔL s increased with increasing h rc and decreased as a function of d b (Table 3b). High d s was associated with increased ΔL s .
The graphical plots showed an increase in ΔL b (Table 3a) towards the crown apex and as a function of branch diameter (Fig. 6a). ΔL b was highest in branches that showed the lowest d s,av values and which had most of their shoots close to the branch base (Fig. 6b).
ΔL s (Table 3b) increased towards the crown apex and decreased as a function of d b (Fig. 7a). High values of d s were associated with increased ΔL s (Fig. 7b).

Model-predicted crown features
The model-predicted trajectories of long-term branch elongation were capable of reproducing the dynamic development of crown profile and the accumulation of branch length and biomass in the course of tree growth, which are features not addressed by typical static crown profile models (Power et al., 2012;Crecente-Campo et al., 2013;Gao et al., 2017). At the same time, the models served as construction rules applicable throughout entire silver birch crowns and they predicted many non-trivial features of crown formation, which previously have been reported in various species and in separate studies. The models reproduced a higher growth rate towards the tree apex in terms of both ΔL b (Goulet et al., 2000;Colombo and Templeton, 2006) and ΔL s (Remphrey et al., 2002, Takahashi et al., 2006, the decrease in ΔL b and ΔL s in the lowest branches (Goulet et al., 2000;Umeki and Kikuzawa, 2000), the effect of branching order (measured here as d s ) on ΔL s (Kozlowski and Ward, 1961) and the higher values in the ΔL b of thick branches in comparisons using equal branch height (Goulet et al., 2000).
Within-crown accumulation of branch biomass, estimated as the total length of the woody branch parts, was closely similar to the growth trajectory based on direct biomass measures (Tahvanainen and   Table 3 Final variables and parameter values ( ± 95 % confidence interval) in the power-law models constructed to describe a) total branch elongation and b) individual long shoot elongation. 36.2 ± 6.0 1.0 ± 0.9 0.9 ± 0.2 −0.6 ± 0.2 Table 4 The Bayesian Information Criterion (BIC) for the corresponding model, and the intercept and slope of the regression between the observed vs. the predicted values for the alternative models of a) total branch elongation (Fig. 2) and b) individual long shoot elongation (Fig. 3). Significance tests for the regression parameters given as * P ≤ 0.05 and *** P ≤ 0.001.

Model performance
The predicted crown profile corresponded to measurements of branch length and was comparable to our previous architectural model (Lintunen et al., 2011). However, the predicted L c of the longest primary branches remained below the approximate 3-m maximum reported for mature silver birch (Ilomäki et al., 2003;Sellin and Kupper, 2006), and observed also in the largest crowns of our data. In part this may reflect the effects of stand density on crown development (Ilomäki et al., 2003), hence potentially on the model parameters as well, but may also reflect the unknown consequences of using the growth measurements of just a single year for model parameterization. Similarly, the largest crowns in our data had also considerably higher L t than that predicted by the model, which suggests the need for caution in applying the model to tree and stand conditions not typical for our data.
The crown profile and the resulting final branch length predicted by our models was also sensitive to the effect of d s , which we estimated simply by assuming a taper model common for all branch positions. Thus, the model precision can probably be improved by using refined values for d s and by re-estimating its associated parameter g2.
Overall, the new models were simpler than our previous model versions and provided reasonably good fit, considering the wide size range and geographical distribution of the sample trees. The amount of variation in growth of birch individuals can be large even in far more strictly defined tree samples (Umeki and Kikuzawa, 2000;Umeki and Seino, 2003).

Modelling processes of crown construction
We consider the dynamic nature of the models as an important step towards incorporating the actual crown construction processes in applications that utilize the structural features of crown shape. An efficient tool in the process is the selection of variables for the final models, which acts as a procedure for identifying key factors that modify branch growth. A striking feature of the final models was that they operated with morphological variables that are measurable by hand, or currently perhaps using terrestrial laser scanning (Raumonen et al., 2015;Sievänen et al., 2018). The predictable behaviour of the models suggests that morphological measurements for estimating the amount of annual growth together with power-law approximation can be successfully used to analyse and construct models that capture many essential features of actual growth processes.
In contrast to purely stochastic tree clones generated from laser scanning data (Potapov et al., 2016;, power-law models can provide insight into physiological processes, because they can easily be refined to include and assess the importance of additional physiological factors. In contrast to a more traditional approach, in which a separate model is used to calculate the amount of resources and empirical growth rules are used to generate crown architecture (Renton et al., 2005b), the power-law approach demonstrated its capability for dynamic generation of realistic crown features on its own.
It was also evident that the simple allometric model, in which the normalization constant was included as a single variable without any spatial information, was inadequate to predict the variability of growth in individual branches. Even though the allometric model reasonably followed the average growth trajectory of branches, it failed to depict the slowing-down of branch growth towards the crown base (Maillette, 1982). Thus, the power-law models provided improved features compared to the simple allometric model, and also served as a tool for Fig. 2. Plots with observed vs. predicted total branch elongation (ΔL b ) using different models. The 1:1 lines are also shown. Fig. 3. Plots with observed vs. predicted individual long shoot length (ΔL s ) using different models. The 1:1 lines are also shown. P. Kaitaniemi, et al. Ecological Modelling 416 (2020) 108900 assessing factors that may influence the values of constituents in allometric equations and their patterns of covariation. The values of the scaling exponent and normalization constant are tightly linked, and various combinations of values may provide acceptable fit with empirical observations (West and West, 2011). Analysing power-law models with or without assuming a constant scaling exponent can be used to reveal information on processes that generate variation in empirically observed scaling relationships (Kaitaniemi and Lintunen, 2008).
A useful future enhancement would be the addition or replacement of driving X i variables using variables that directly indicate the resources available for crown development. A more explicit consideration of the branching topology and the sequence of events during crown and stand development may also be beneficial, because various mechanisms can contribute to growth at different crown positions and at different times of tree ontogeny. At the crown apex, the availability of light may guide upward growth sufficient for a tree to persist in competition, although at some point hydraulic constraints will limit the increase in height (Ishii et al., 2008). At lower crown positions, dynamically changing differences in relative shading (Henriksson, 2001), amount of growing space (Jones and Harper, 1987), branch orientation in relation to the sun (Stoll and Schmid, 1998), GV, and timing of events (Duchesneau et al., 2001) may be more influential and give rise to other typical responses, such as crown asymmetry. The costs of maintenance are likely to increase as branch size increases (Cannell and Morgan, 1989;Spatz and Bruechert, 2000), and the longest and most vigorous branches can also become subject to mechanical abrasion with neighbouring trees, both setting limits to branch growth and GV (Hajek et al., 2015;Loehle, 2016). Since neighbouring trees and other biotic factors undergo simultaneous changes, crown structure is likely to be rarely at the fully optimal state.
Since we did not detect the effect of neighbour distance on branch growth, the distance to neighbouring trees may have occurred over a range in which a change in distance itself does not explain crown expansion (Simard and Zimonick, 2005). The stand densities were relatively low , and extensive mechanical contacts generating prominent crown asymmetry were not anticipated or observed. We also used an average measure of distance for all the basal branches throughout the crown, whereas mechanical contacts may be prominent only in specific compass directions and in the longest branches (Hajek et al., 2015).
The list of explanatory variables in the final models included indirect factors such as d s , h rc and l rb, av , whereas some direct factors were omitted, such as PAR and N area , which indicate potential photosynthetic capacity within the crown. The exclusion of PAR and N area suggests that the indirect variables defining mainly the relative position of the branches and shoots within the crown were sufficient to capture the effect of PAR and N area . There are several examples of positional effects in addition to light that can influence growth responses, and the use of d s , h rc and l rb, av may have indicated the favourability of growth positions relative to the structural organization of the entire crown. For example, differences in the relative levels of shading instead of absolute differences in PAR may greatly modify the allocation of growth (Henriksson, 2001;Dong et al., 2015), while differences in height or GV Fig. 4. a) Growth trajectories for the total length of all woody branch parts predicted by the models Table 3a (black line), eq. (5) (dotted line) and eq. (6) (short dashes). The symbols indicate average observed total branch length in three tree age classes (white circle is below 10, triangle is 10-20, and black circle is above 20 years). Branch age was used to estimate a standardized h rc for the observed branch length in different tree age classes (see Appendix A). b) Cumulative total length of all woody branch parts in relation to the maximum total branch length predicted by the models in a), and compared with the model of proportional cumulative biomass (Tahvanainen and Forss, 2008) at different values of h rc (long dashes). A new branch is initiated at lower left and reaches its maximum size at upper right.  Table 3b (black line) and eq. (7) (dotted line), illustrated as the distance from the branch tip to the stem, together with an approximate crown profile (dashed line) estimated from the trajectory of axis length predicted by model Table 3b. Growth of a single primary branch axis was predicted as its h rc decreases during stem elongation. A 90°branching angle was used in the plots for growth trajectories, and a 45°branching angle in the plot of crown profile. The symbols indicate average observed primary branch axis length in three tree age classes (white circle is below 10, triangle is 10-20, and black circle is above 20 years). Branch age was used to estimate a standardized h rc for the observed branch length in different tree age classes as explained in Appendix A.
among branches may affect growth independently of light (Goulet et al., 2000, Osada et al., 2014, and mechanisms of apical control may also contribute to growth allocation (Duchesneau et al., 2001, Palubicki et al., 2009. The use of d s , h rc and l rb, av as indirect explanatory variables in the models can be considered as a shortcut to cover multiple possible configurations of internal organization, although there may remain the need for more precise definition through the specification and inclusion of additional factors X i in the models. However, many alternative trait configurations may provide plants with equal performance in terms of growth or other measures of success (Delagrange et al., 2004;Hubbell, 2006;Marks and Lechowicz, 2006;Kaitaniemi, 2007), thus even a simplified model can be operationally efficient for many purposes (Rosindell et al., 2012).
Some important factors were omitted from the models for reasons of simplicity or lack of sufficient data. For example, temperature, precipitation and site fertility greatly influence tree growth (Niinemets and Lukjanova, 2003;Dewar et al., 2009), and in models where prediction at multiple locations and for several years is required, they are clearly candidates to be included with their own X i as driving variables to indicate the availability of resources for growth. Investment in reproduction or losses to herbivory could be included as an efflux that removes resources from growth. We also ignored the effects of neighbouring species and competition indices, which we previously reported as being influential for silver birch growth and crown construction . Instead, they were considered through the use of PAR and N area , both of which increase with the presence of silver birch as the dominant neighbouring species in comparison to Scots pine neighbours with equal height . The positional variables d s , h rc and l rb may have incorporated the direct effects of PAR and N area .

Conclusions
We conclude that the power-law approach shows good potential for modelling and analysing crown development at the level of 3D crown structure, using variables that capture features of both structure and functioning of crowns. The models can provide improved precision regarding allocation of growth compared with classical statistical models or simple allometric estimates of growth. Even further precision may be obtained by including resource availability and identifying the consequences of processes associated with detailed branching topology and spatial interactions among trees during stand development.

Declaration of Competing Interest
None. Fig. 6. Effect of selected variables in the model in Table 3a on total branch elongation (ΔL b ). Other variables were set to constant values shown inside the figures.  Table 3b on individual long shoot elongation (ΔLs). Other variables were set to constant values shown inside the figures. P. Kaitaniemi, et al. Ecological Modelling 416 (2020)