Determining an Accurate and Cost-Effective Individual Height-Diameter Model for Mongolian Pine on Sandy Land

: Height-diameter (H-D) models are important tools for forest management practice. Sandy Mongolian pine plantations ( Pinus sylvestris var. mongolica ) are a major component of the Three-North Afforestation Shelterbelt in Northern China. However, few H-D models are available for Mongolian pine plantations. In this paper we compared different equations found in the literature for predicting tree height, using diameter at breast height and additional stand-level predictor variables. We tested if the additional stand-level predictor variable is necessary to produce more accurate results. The dominant height was used as a stand-level predictor variable to describe the variation of the H-D relationship among plots. We found that the basic mixed-effects H-D model provided a similar predictive accuracy as the generalized mixed-effects H-D model. Moreover, it had the advantage of reducing the sampling effort. The basic mixed-effects H-D model calibration, in which the heights of the two thickest trees in the plot were included to calibrate the random effects, resulted in accurate and reliable individual tree height estimations. Thus, the basic mixed-effects H-D model with the above-described calibration design can be an accurate and cost-effective solution for estimating the heights of Mongolian pine trees in northern China.


Introduction
Shelterbelt establishments are an effective solution in preventing natural disturbances and hazards such as soil erosion, avalanches or landslides [1]. The Three-North Shelter Forest Program is the largest artificial forest shelter program in the world [2]. This shelter forest program plays a vital role in carbon sequestration and sand fixation, soil conservation, wood production and supply for the local communities [3]. By 2018, a forest area plantation of 3.01 × 10 7 hm 2 had remained from the Three-North Shelter Forest Program [3]. Sandy Mongolian pine (Pinus sylvestris var. mongolica) is one of the most important and common species used for afforestation in the Three-North Shelter Forest Program. The area of sandy Mongolian pine plantations in northern China exceeds 7 × 10 5 hm 2 [4].
Tree height and diameter at breast height are two elementary input variables used in forest management practice [5,6], with tree height being an input variable needed in

Ssampling Design and Data Collection
For this research thirty-five temporary sample plots of 400 m 2 (20 m × 20 m) were established in Mongolian pure pine stands. The sampling plots cover different stand ages and stand structures of the Mongolian pine plantations. The stand age ranges from 13 to 62 years, and includes young, half-mature, mature, and old forests. For each stand age the sampling plots include different stand densities. The detailed sampling workflow can be found in [29,30]. Stand age was obtained for each plot based on the afforestation records. Stand density was registered as the counts of living trees. For all living and healthy trees in each plot, the diameters at breast height (D: 1.3 m above ground) were measured using diameter tape with a precision of 0.1 cm; tree height (H) of all living trees was measured using an infrared dendrometer (Criterion™ RD1000) with a precision of 0.1 m.
Only the trees with D ≥ 5 cm were used in this analysis. A total of 1070 paired height-diameter measurements were chosen for the model development. This dataset was used for the parameter estimation and model calibration. The distributions of the height and diameter at breast measurements are illustrated in Figure 1. mean annual temperature is approximately 5.3 °C, and the multiyear annual precipitation ranges from 300 to 500 mm [29]. The major local vegetation consists of P. tabuliformis, Agriopyllum squarrosum, Salix gordejevii, Artemisia halodendron, and Calamagrostis epigeios [30].

Ssampling Design and Data Collection
For this research thirty-five temporary sample plots of 400 m 2 (20 m × 20 m) were established in Mongolian pure pine stands. The sampling plots cover different stand ages and stand structures of the Mongolian pine plantations. The stand age ranges from 13 to 62 years, and includes young, half-mature, mature, and old forests. For each stand age the sampling plots include different stand densities. The detailed sampling workflow can be found in [29,30]. Stand age was obtained for each plot based on the afforestation records. Stand density was registered as the counts of living trees. For all living and healthy trees in each plot, the diameters at breast height (D: 1.3 m above ground) were measured using diameter tape with a precision of 0.1 cm; tree height (H) of all living trees was measured using an infrared dendrometer (Criterion™ RD1000) with a precision of 0.1 m.
Only the trees with D ≥ 5 cm were used in this analysis. A total of 1070 paired heightdiameter measurements were chosen for the model development. This dataset was used for the parameter estimation and model calibration. The distributions of the height and diameter at breast measurements are illustrated in Figure 1.

Basic Model Selection
As reported in previous studies, nonlinear curves are more suitable than linear ones for representing H-D relationships [15,16]. Following these findings and examining the distributions of our H-D dataset, we chose 8 nonlinear candidate H-D models (Table 1). To ensure parsimony and convergence for each plot, only models with two parameters were selected. The suitability of all candidate models in predicting height was validated in previous research papers [12,31]. The models' performances were compared using the coefficient of determination (R 2 ), the mean absolute error (MAE), and the root mean squared error (RMSE). The expressions of R 2 , MAE, and RMSE are shown in Equations (9)-(11): Figure 1. Distributions of the diameter at breast height (a) and height (b) as well as plot-specific fitted height-diameter lines (c). Plot-specific line was obtained by fitting height-diameter (H-D) relationships in each plot using the best basic model determined below.

Basic Model Selection
As reported in previous studies, nonlinear curves are more suitable than linear ones for representing H-D relationships [15,16]. Following these findings and examining the distributions of our H-D dataset, we chose 8 nonlinear candidate H-D models (Table 1). To ensure parsimony and convergence for each plot, only models with two parameters were selected. The suitability of all candidate models in predicting height was validated in previous research papers [12,31]. The models' performances were compared using the coefficient of determination (R 2 ), the mean absolute error (MAE), and the root mean squared error (RMSE). The expressions of R 2 , MAE, and RMSE are shown in Equations (9)-(11): H is the height, D is the diameter at breast height, a and b are the model coefficients, ε is the error term.
where R 2 is the coefficient of determination; MAE is the mean absolute error; RMSE is the root mean squared error; H ij andĤ ij are the observed and predicted heights of tree j on plot i, respectively; H is the average of the observed heights; n i is the number of trees in the plot i; m is the total number of sample plots; n is the total number of trees.

H-D Generalized Model
Additional stand-level predictor variables were added to the best basic H-D model in order to develop the generalized H-D model. Among the stand-level predictor variables considered were stand age (Age), stand dominant height (DH), stand basal area (BA), stand density (N), relative spacing (RS), stand quadratic mean diameter (Dg), stand arithmetic mean diameter (AMD), maximum diameter (Dmax), minimum diameter (Dmin), diameter range per plot (Dr) ( Table 2). In the present work, DH was defined as the mean height of the 100 largest-diameter trees per hectare [10], as there are no available dominant height models developed for this species. To obtain a generalized model with a high accuracy and an easier biological interpretation, a two-stage reparameterization method was used [38]. In the first step linear regression was conducted using the estimated coefficients for each plot as response variables and the stand-level predictor variables as independent variables. The variables with variance inflation factor (VIF) > 5 were removed to eliminate multicollinearity. Finally, the remaining and most significant variables were used to develop the generalized H-D model.

Nonlinear Mixed-Effects Model
Based on the OLS basic and generalized models we developed nonlinear mixed-effects models taking into account the hierarchical structure of the data. Additional random parameters associated with the equation's coefficients were considered. The Akaike information criterion (AIC) and the Bayesian information criterion (BIC) were used to determine the prediction gain by including additional random effects in the following general form of the mixed-effects model [28]: where x i and y i are the response and predictor variables, respectively; f is a nonlinear function between the response and predictor variables; β is a q × 1 fixed parameter vector; u i is a random effects parameter for plot i and is assumed to be distributed normally with variance-covariance matrix Ψ, i.e., u i~N (0, Ψ); ε i is error term and is assumed to be normally distributed with the within-sample-plot variance-covariance matrix for R i , i.e., ε i~N (0, R i ).
The expression of R i is as follows [39]: where σ 2 is the scale; G i is a diagonal matrix describing the heteroskedasticity; and Γ i is an autocorrelation structure of error and is an identity matrix in this study. Three variance functions: exponential, power, and constant plus power, with D as the predictor, were compared in accounting for the heteroscedasticity. These formulas are shown below: var where σ 2 is the scale, and δ is the parameter to be estimated.

Parameter Estimation
The parameters of the basic and generalized H-D models were first estimated with OLS method using the nls function in R software [40]. Parameters of the mixed-effects models were estimated in the mixed-effects modelling framework using the nlme function and the restricted maximum likelihood (REML) method in the nlme package in R software [40,41].

Evaluation of Model Prediction Performance and Appropriate Calibration Design
The mixed-effects models were used for the following types of height predictions in the stand of the entire data: (1) Fixed-effects prediction of the mixed-effects model, where only the fixed part was used as follows: where H i and x i are the height and predictor vectors of plot i;β is the vector of estimated fixed parameters; ε i is error term.
(2) Random-effects prediction of the mixed-effects model for a specific plot. The random-effects predictions require the random-effects parameters u i to be calculated with the following Equation [42]: whereû i is the estimated random effects parameters;ψ andR i are the estimations of Ψ and R i , respectively;Ẑi is the design matrix for the random effects; andẐ T i is the transpose matrix ofẐ i .
In this paper,û i was calculated using the empirical best linear unbiased prediction (EBLUP) method [43]. The calibration design involved different sampling strategies. The tree size categories sampled include the thinnest, medium-sized and thickest trees. A total of 4 designs with 47 combinations were evaluated in the random-effects predictions.
(i) One-six (in order) trees from the same diameter size category were selected. (ii) One-three (in order) trees were chosen separately from two diameter size categories. (iii) One tree from each of the three diameter size categories was selected separately. (iv) Two trees were selected separately from each of the three diameter size categories.
In order to determine the reliable and most accurate calibration we first used the paired Wilcoxon rank-sum test to test which calibration design leads to an unsignificant difference between predicted and observed heights. Then, the best calibrations obtained for each number of trees sampled were further compared using MAE and RMSE to determine the best calibration solution for the two mixed-effects models developed.

Basic and Generalized Fixed Model
Although the performances did not drastically differ among the eight models (Table 3), model 4 (M4) had the highest R 2 , and the lowest MAE and RMSE values. In addition, M4 could also fit each plot in our dataset ( Figure 1c). Thus, M4 was selected as the basic model for further model development. The majority of stand-level predictor variables were significantly correlated with the coefficient β 1 in M4 (Equation (19)), while no stand-level predictor variable was significantly correlated with the parameter β 2 of Equation (19) (Figure 2). Finally, only the DH was retained in the generalized H-D model (Equation (20)) according to the two-step linear regression and correlation ( Figure 3). The generalized model obtained significantly better goodness-of-fit (p < 0.0001): where H ij and D ij are the H and D of tree j in plot i, respectively; β 1 , β 2 , β 3 are fixed coefficients; DH i is the dominant height for sample plot i; and ε ij is the error.

Mixed-Effects Models
The final formulas for the basic and generalized mixed-effects models are presented in Equations (21) and (22), respectively. The power variance function showed better performance than the other two functions in removing the heteroscedasticity of basic and generalized mixed-effects models (Table S1, Figure 4). where Hij and Dij are the H and D of tree j in plot i, respectively; β1, β2, β3 are cients; DHi is the dominant height for sample plot i; and εij is the error.   where Hij and Dij are the H and D of tree j in plot i, respectively; β1, β2, β3 a cients; DHi is the dominant height for sample plot i; and εij is the error.    formance than the other two functions in removing the heteroscedasticity of basic and generalized mixed-effects models (Table S1, Figure 4).  The generalized mixed-effects model had lower AIC and BIC values than the basic mixed-effects model ( Table 4) and was also significantly different from the basic mixedeffects model (p < 0.0001). The random-effects values of the basic mixed-effects model were larger than those of generalized mixed-effects model, as the standard deviation of the basic model random-effects was ten times larger than that of the generalized mixedeffects model. The DH explains the high variety of H-D relationships as random-effects do (Figure 5a). In addition, for the trees with same diameter, the tree height increased with the increase in the stand DH (Figure 5b). The basic mixed-effects model, however, had an almost similar (even better) performance as the generalized mixed-effects model according to the R 2 , MAE, and RMSE (Table 4). In addition, the two mixed-effects model had similar distributions of the standardized residuals ( Figure 4).  The generalized mixed-effects model had lower AIC and BIC values than the basic mixed-effects model ( Table 4) and was also significantly different from the basic mixedeffects model (p < 0.0001). The random-effects values of the basic mixed-effects model were larger than those of generalized mixed-effects model, as the standard deviation of the basic model random-effects was ten times larger than that of the generalized mixed-effects model. The DH explains the high variety of H-D relationships as random-effects do (Figure 5a). In addition, for the trees with same diameter, the tree height increased with the increase in the stand DH (Figure 5b). The basic mixed-effects model, however, had an almost similar (even better) performance as the generalized mixed-effects model according to the R 2 , MAE, and RMSE (Table 4). In addition, the two mixed-effects model had similar distributions of the standardized residuals (Figure 4).

Mixed-Effects H-D Model Calibration
The random-effects predictions had lower MAE and RMSE values than the fixed predictions and were closer to the observed height values (Figures 6a,b and 7a,b, Table S2). The generalized mixed-effects model had better prediction accuracy than the basic mixedeffects model under the majority of calibration designs with lower MAE and RMSE (Figure 6a,b, Table S2). In addition, the number and the diameter size of measured trees had an obvious effect on the prediction accuracy of the basic mixed-effects model; however, this was not the case for the generalized H-D model (Table S2). The calibration accuracy increased with the increase in prior height measurements; nevertheless, the accuracy increased negligibly (<5 cm) when the number of trees used for calibration was higher than two (Table S2).

Mixed-Effects H-D Model Calibration
The random-effects predictions had lower MAE and RMSE values than the fixed predictions and were closer to the observed height values (Figures 6a,b and 7a,b, Table S2). The generalized mixed-effects model had better prediction accuracy than the basic mixedeffects model under the majority of calibration designs with lower MAE and RMSE (Figure 6a,b, Table S2). In addition, the number and the diameter size of measured trees had an obvious effect on the prediction accuracy of the basic mixed-effects model; however, this was not the case for the generalized H-D model (Table S2). The calibration accuracy increased with the increase in prior height measurements; nevertheless, the accuracy increased negligibly (<5 cm) when the number of trees used for calibration was higher than two (Table S2).

Mixed-Effects H-D Model Calibration
The random-effects predictions had lower MAE and RMSE values than the fixed predictions and were closer to the observed height values (Figures 6a,b and 7a,b, Table S2). The generalized mixed-effects model had better prediction accuracy than the basic mixedeffects model under the majority of calibration designs with lower MAE and RMSE (Figure 6a,b, Table S2). In addition, the number and the diameter size of measured trees had an obvious effect on the prediction accuracy of the basic mixed-effects model; however, this was not the case for the generalized H-D model (Table S2). The calibration accuracy increased with the increase in prior height measurements; nevertheless, the accuracy increased negligibly (<5 cm) when the number of trees used for calibration was higher than two (Table S2). Although the best design with different tree number showed that increasing height measurements improved the prediction accuracy of the basic mixed-effects model, the improvement of accuracy was negligible (<5 cm) when the number of measured trees was Although the best design with different tree number showed that increasing height measurements improved the prediction accuracy of the basic mixed-effects model, the improvement of accuracy was negligible (<5 cm) when the number of measured trees was more than two (Table S2). In addition, the difference in random-effects prediction performances between the two mixed-effects models was very small when more than two tree heights were available (Figure 6a,b).
To obtain an increased prediction accuracy with as few prior tree height measurements as possible, the basic mixed-effects model required the two thickest trees' heights, and the generalized mixed-effects model required one medium tree height. The two mixed-effects models had similar predicted results based on the above-mentioned designs (Figure 7). heights were available (Figure 6a,b).
To obtain an increased prediction accuracy with as few prior tree height measurements as possible, the basic mixed-effects model required the two thickest trees' heights, and the generalized mixed-effects model required one medium tree height. The two mixed-effects models had similar predicted results based on the above-mentioned designs (Figure 7).

Basic and Generalized H-D Models
Our study's aim was to build an appropriate H-D model for the Mongolian pine stands in Zhanggutai, where the oldest introduced sandy Mongolian pine plantations in China are located [44]. We developed a mixed-effects H-D model and determined an economical method and a simple but effective calibration design for estimating the heights of Mongolian pines to aid forest management planning [6,21].
We used equations with only two coefficients due to the nonconvergence associated with three-parameter models when fitting each plot. Nonconvergence issues have been reported in several studies that aimed to develop nonlinear H-D models [6,27,45]. Among the eight candidate models, the model of Schumacher [35] was selected as the basic H-D model because it had the highest accuracy compared to the other seven models tested. This model has also chosen as a basic mathematical form, suitable for further improvement in similar research studies [7,21,27].
Developing generalized H-D models is an effective solution to account for the variations in H-D relationships among plots. Generalized H-D models can broaden the applications of such predictive tools. In this work, we used a common method to link the parameters and stand-level predictor variables and to develop the generalized H-D model [38]. This method has been widely employed to build generalized H-D models [6,7,15]. Among the tested stand-level predictor variables, only the DH was determined to be necessary for developing the generalized H-D model. DH positively and significantly correlated with the asymptote coefficient of the candidate model. Our results are consistent with those of many previous studies [5,17,21,46] that used dominant height as an additional stand-level predictor variable associated with the asymptote coefficient. This could be expected, as DH is a measure of the stand's maximum height potential, which is usually

Basic and Generalized H-D Models
Our study's aim was to build an appropriate H-D model for the Mongolian pine stands in Zhanggutai, where the oldest introduced sandy Mongolian pine plantations in China are located [44]. We developed a mixed-effects H-D model and determined an economical method and a simple but effective calibration design for estimating the heights of Mongolian pines to aid forest management planning [6,21].
We used equations with only two coefficients due to the nonconvergence associated with three-parameter models when fitting each plot. Nonconvergence issues have been reported in several studies that aimed to develop nonlinear H-D models [6,27,45]. Among the eight candidate models, the model of Schumacher [35] was selected as the basic H-D model because it had the highest accuracy compared to the other seven models tested. This model has also chosen as a basic mathematical form, suitable for further improvement in similar research studies [7,21,27].
Developing generalized H-D models is an effective solution to account for the variations in H-D relationships among plots. Generalized H-D models can broaden the applications of such predictive tools. In this work, we used a common method to link the parameters and stand-level predictor variables and to develop the generalized H-D model [38]. This method has been widely employed to build generalized H-D models [6,7,15]. Among the tested stand-level predictor variables, only the DH was determined to be necessary for developing the generalized H-D model. DH positively and significantly correlated with the asymptote coefficient of the candidate model. Our results are consistent with those of many previous studies [5,17,21,46] that used dominant height as an additional stand-level predictor variable associated with the asymptote coefficient. This could be expected, as DH is a measure of the stand's maximum height potential, which is usually associated with the site productivity and the site index [47]. DH is a result of the competition process present in the stand, the stand density [17], and the stand quality [5,46]. Although significant correlation between the coefficients and the stand-level predictor variables such as the Age, the N and the BA was detected, adding other stand-level predictor variables did not increase the model performances significantly. In this work, DH was significantly correlated with other stand-level predictor variables except the Dr. One of the reasons why the stand-level predictor variables that were significantly correlated with the coefficient of the basic model did not significantly improve the H-D model is the multicollinearity. Finally, the generalized H-D model containing only DH as an additional stand-level predictor variable was chosen as the best generalized H-D model for the Mongolian pine stands in the study area.

Mixed-Effects H-D Models
Mixed-effects models are more flexible and have a better predictive accuracy in modeling H-D relationships. Furthermore, they can considerably improve the model prediction accuracy through different calibration scenarios [17,24]. In this study, the basic mixed-effect model includes only one random-effect parameter. In addition, the randomeffect was included in the asymptote coefficient, which indicated that the maximum height had a high variation between stands [17,25].
Generalized mixed-effects models containing stand-level predictor variables that can account for the variability among stands generally achieve a better performance in fitting and prediction [27]. The DH stand-level predictor variable significantly improved the basic model according to the AIC and the BIC statistics. In addition, the generalized mixed-effects model also had better predictive accuracy under the majority of calibration designs, which is consistent with our first hypothesis. Nevertheless, the MAE and RMSE statistics did not improve based on the results of fitting and the best calibration design. Similar results were described by Zang et al. (2016), where their basic mixed-effects H-D model for larch plantations in northern and northeastern China provided better RMSE and MAE statistics than the generalized mixed-effects model [25]. Furthermore, Huang et al. (2009) found that the basic mixed-effects model performed as well as or better than the generalized mixed-effects model for aspen grown in boreal mixed wood forests in Alberta, which is similar to our study in that the inclusion of other stand-level predictor variables in the model did not produce more accurate predictions [5]. A possible explanation for why the generalized mixed-effects H-D model does not provide a better accuracy compared to the basic mixed-effects model is related to the capacity of the random-effect parameter to explain the various H-D relationships among stands. Our results support this theory, since both the stand-level predictor variable and the random effects were introduced alongside the asymptote coefficient of the model, and the random effects of the basic mixed-effects model were significantly larger than those of the generalized mixed-effects model. In addition, the estimated random-effects of the basic mixed-effects model were strongly and significantly correlated with the stand-level predictor variable DH. However, generalized models provide a better biological representation of the process studied and are helpful for large datasets [25,48]. Thus, whether or not the generalized mixed-effects model is still needed to predict the height for the sandy Mongolian pine plantations needs further investigation. Furthermore, DH models are not yet available for this species as far as we know. This would make the generalized model even more difficult to apply as the field work would involve measuring the height of the tallest 100 trees or estimating DH based on a sample plot. Considering that similar results can be obtained with the basic mixed-effects model by predicting the random effect with a reduced number of trees measured on the field, we recommend the use of the basic mixed-effects model to reduce the necessary field work.

Calibration for Random-Effects Prediction
The most important reason for developing H-D models is to estimate tree heights using tree diameters at breast height for new stands where these measurements are missing. The mixed-effects framework allows for the calibration of the fixed effects in order to obtain more accurate predictions than the ones obtained using the population mean coefficients. The calibration design (which mainly includes the number and the size of trees) has an obvious effect on the predictive accuracy of the model [49][50][51]. Furthermore, the appropriate calibration design varies between forest types and species [6,7,17]. To determine the most appropriate prediction strategy, 47 calibration combinations were analyzed separately for the generalized and basic mixed-effects H-D models. As we sug-gested in our second hypothesis, the MAE and the RMSE decreased with the increase in the number of trees. However, the decrease in the MAE and the RMSE was different for each of the two mixed-effects models. The MAE and the RMSE for the generalized mixed-effects H-D model fluctuated slightly when changing the sampling design, similar to Crecente-Campo et al. (2014) [52]. Unlike the generalized mixed-effects H-D model, the basic mixed-effects H-D model requires less field work effort and costs for randomeffects calibration. Moreover, the basic mixed-effects H-D model does not require the extra DH measurement in addition to the D measurements. Moreover, the basic mixedeffects H-D model required only two height measurements for the calculation of random parameters and yielded similar MAE and RMSE when compared with the generalized mixed-effects model.
Unexpectedly, the third hypothesis was not confirmed for our dataset: sampling trees from different diameter classes did not lead to better calibration results. In this study, we found that for the basic mixed-effects H-D model only the two thickest tree heights are needed in order to calibrate the model and to produce reliable and labor-saving predictions. In contrast to our findings, Bronisz and Mehtätalo (2020) reported that using the extreme trees (the thinnest and the thickest) in Betula pendula stands led to better performances than any other analyzed sampling strategy [7]. Our results are close to those obtained by Ciceu et al. (2020), who reported that the diameter at breast height of medium and thickest trees was the best for model calibration for Picea abies in mixed uneven-aged stands [6].
Our basic mixed-effects model calibration results can be explained based on the strong correlation between the DH and the asymptote coefficient. The heights of the two largest trees serve as an estimation of the dominant height, thus replacing the importance of this variable in the H-D relationship between stands. Thus, using the thickest trees provided more additional information for calibrating the basic mixed-effects model than the information brought by the median tree height in the generalized H-D model.
Following the above-mentioned results, we recommend sampling the two thickest trees per plot for a practical height prediction. However, our study was only conducted in Mongolian pine plantations on sandy land in northwest Liaoning Province of China, as there was no available independent dataset for model validation in other regions. These results should be carefully validated when the region is changed.

Conclusions
Basic and generalized H-D models were obtained for the Mongolian pine plantations on sandy land in northern China. We found that the Schumacher [37] model had the best performance in modeling local H-D relationships relative to the other basic H-D models found in the literature. The DH was found to significantly describe the H-D relationships among plots.
We found that the basic mixed-effects H-D model produces similar prediction to the generalized model but with less sampling effort, which is more suitable for practical applications. Thus, the basic mixed-effects H-D model can be a solution for easily estimating individual tree heights in Mongolian pine plantations where DH measurements are not available. Furthermore, an increased accuracy could be obtained by calibrating the random effects using additional height measurements of the two thickest trees.
The use of this method would reduce the necessary field work which would otherwise be needed to determine the dominant height of the stand by measuring only two heights for random effects calibration.