Height-diameter Models Research Articles

Height-diameter models: a comprehensive review with new insights on relationships to generalized linear models and differential equations

Forest scientists use regression models widely, particularly for height-diameter modeling. These models offer several benefits for estimating height in homogeneous or non-homogeneous stands. The original models for height estimation based on diameter at breast height have been extended to include other variables, thanks to technological advancements. The purpose of this article is to provide a literature review using the methodology outlined by Cervo and Bervian (2011), providing helpful information to forest biometricians in selecting a height-diameter model that utilizes historical data. The models can be classified into four main groups and extended to include other covariates besides diameter at breast height. Many of the models used data transformation but results showed that with the exception of one group (nonlinear models), all other models can be considered a generalized linear model, with corresponding linear predictors and link functions. The paper also discusses the generation process of these models, the species to which they are commonly applied, and how they can be created using general ordinary differential equations.

Tree Height–Diameter Model of Natural Coniferous and Broad-Leaved Mixed Forests Based on Random Forest Method and Nonlinear Mixed-Effects Method in Jilin Province, China

Objective: The purpose of this article was to use the Random Forest method and nonlinear mixed-effects method to develop a model for determining tree height–diameter at breast height (DBH) for a natural coniferous and broad-leaved mixed forest in Jilin Province and to compare the advantages and disadvantages of the two methods to provide a basis for forest management practice. Method: Based on the Chinese national forest inventory data, the Random Forest method and nonlinear mixed-effects method were used to develop a tree height–DBH model for a natural coniferous and broad-leaved mixed forest in Jilin Province. Results: The Random Forest method performed well on both the fitting set and validation set, with an R2 of 0.970, MAE of 0.605, and RMSE of 0.796 for the fitting set and R2 of 0.801, MAE of 1.44 m, and RMSE of 1.881 m for the validation set. Compared with the nonlinear mixed-effects method, the Random Forest model improved R2 by 33.83%, while the MAE and RMSE decreased by 67.74% and 66.44%, respectively, in the fitting set; the Random Forest model improved R2 by 9.88%, while the MAE and RMSE decreased by 14.38% and 12.05%, respectively, in the validation set. Conclusions: The tree height–DBH model constructed based on the Random Forest method had higher prediction accuracy for a natural coniferous and broad-leaved mixed forest in Jilin Province and had stronger adaptability for higher-dimensional data, which can be used for tree height prediction in the study area.

Modelling Differently Defined Dominant Stand Diameters of Monospecific Forest Plantations

Quadratic mean diameter is a widely used stand parameter present in the stand inventory summaries, while the top stand diameter is rarely reported in the literature, mainly in relation to dominant stand height. Since the dominant stand height is usually determined from the tree height-diameter curve of the stand, it is important how the top tree assemblage, used to estimate dominant diameter, is defined. The main objective of our study was to assess the bias between differently defined dominant diameter estimates for monospecific plantations of various species, to model the dominant diameter as a function of quadratic mean diameter and other relevant stand variables, and to estimate its goodness-of-fit in predicting dominant diameter and dominant height.We used data records gathered in sample plots in monospecific plantations of four tree species: Scots pine, Black pine, black locust and hybrid black poplar. We calculated the quadratic and arithmetic mean diameters of the 20% thickest trees in the plots, and the quadratic and arithmetic mean diameters of the trees, whose number corresponded to the 100 thickest trees per hectare. For each dataset, we analyzed the range and the distribution of the relative deviations calculated for each pair of dominant diameter estimates. For the Black pine plantations, regression models were developed for the two dominant diameter definitions, whose values differed most. Their goodness-of-fit was assessed from model efficiency and error statistics. The same model derivation procedure, applied to the Scots pine data, was followed by substitution of the predicted dominant diameter into a height-diameter model to assess the goodness-of-fit of the dominant height predictions.The differences between the arithmetic and quadratic means, estimated from the same subsample of trees, did not exceed 2% in all cases. However, dominant stand diameters calculated as averages of differently defined largest tree collectives differed by as much as 35%. Regardless of its definition, the dominant stand diameter was adequately predicted by a function of the quadratic mean diameter alone or considering stand basal area as a second predictor. The models showed very good accuracy of model efficiency above 0.92, average absolute error below 8%, with 90% of the relative errors less than 15%. The predicted dominant diameter value can be used in a height-diameter model to estimate with confidence the dominant stand height of a monospecific forest plantation, allowing the forecast of the stand attributes based on dominant trees when only average stand variables are known.

A mixed-effects height-diameter model for Pinus douglasiana Martínez in temperate forests of Jalisco, Mexico

predicting tree-height of Pinus douglasiana in the natural forests of Jalisco, Mexico. Design/methodology/approach: For this study, we utilized 2,921 pairs of tree-height measurements collected from 65 permanent plots (each 50x50 m) in the study area. For each plot, we estimated the tree density as the number of trees per hectare (N, ha-1), the basal area per hectare (G, m2 ha-1) and the quadratic mean diameter (dg, cm). Subsequently, we tested several models from the literature and developed a generalized mixed-effects model for tree height prediction. Results: The Gompertz base model outperformed the other local models, achieving an R² of 0.75 and an RMSE of 3.13. Including the stand variables (N, G, and dg) and incorporating random effects significantly improved the model fit, resulting in an R² of 0.89 and an RMSE of 2.09 m. Calibration and validation steps revealed that selecting the three thickest trees is effective for estimating random effects in new plots or stands, with the RMSE and R² being 2.59 and 0.83, respectively. Limitations on study/implications: The present model can be applied in areas where P. douglasiana is naturally distributed. However, it is advisable to calibrate the model using a sub-sample to achieve more accurate predictions of tree heights in new plots or areas. Findings/conclusions: The Gompertz function was used as the base function to develop the final generalized mixed-effects model for predicting the height of Pinus douglasiana. The inclusion of stand variables (N, G, and dg) along with random effects improved the model fit. This model represents a new and more accurate tool for predicting the height of P. douglasiana in areas where it is naturally distributed.

Ecoregional height–diameter models for Scots pine in Turkiye

Ecoregion-based height-diameter models were developed in the present study for Scots pine (Pinus sylvestris L.) stands in Turkiye and included several ecological factors derived from a pre-existing ecoregional classification system. The data were obtained from 2831 sample trees in 292 sample plots. Ten generalized height–diameter models were developed, and the best model (HD10) was selected according to statistical criteria. Then, nonlinear mixed-effects modeling was applied to the best model. The R2 for the generalized height‒diameter model (Richards function) modified by Sharma and Parton is 0.951, and the final model included number of trees, dominant height, and diameter at breast height, with a random parameter associated with each ecoregion attached to the inverse of the mean basal area. The full model predictions using the nonlinear mixed-effects model and the reduced model (HD10) predictions were compared using the nonlinear sum of extra squares test, which revealed significant differences between ecoregions; ecoregion-based height–diameter models were thus found to be suitable to use. In addition, using these models in appropriate ecoregions was very important for achieving reliable predictions with low prediction errors.

Improving plot-level above ground biomass estimation in tropical Indian forests

Accurate plot-level estimates of above ground biomass (AGB) are crucial for reducing uncertainty in spatial AGB quantification, a key requirement for Reducing Emissions from Deforestation and Forest Degradation framework (REDD+) and achieving Intended Nationally Determined Contributions (INDC) goals. This study explores the impact of plot size, selection and use of height-diameter (H–D) model, and allometric models on plot-level AGB estimation in tropical Indian forests. We analyze a dataset of 8179 tree measurements from 51 one-hectare plots across four tropical sites to investigate these factors and improve estimation accuracy. Our findings highlight the importance of locally derived H–D models for superior accuracy. We propose two H–D models for improved plot-level AGB estimation accuracy in tropical Indian forests: a general model with a residual standard error (RSE) of 4.18 m and a forest type-specific model with a RSE of 3.65 m. These models outperform global models, which significantly underestimate tree height by up to 51.3% and overestimate by 43.4% across our sites. The developed H–D models were evaluated using spatial cross-validation to ensure broader applicability across Indian tropical forests. Further, we evaluated the impact of the choice of the allometric model on plot-level AGB estimation using the prevalent models (local destructive models by the Forest Survey of India and global pantropical models). Our results indicate minimal variability between the local destructive models and the pantropical model when height measurements are derived from this study's developed H–D models. We have also investigated the effect of plot size on plot-level AGB estimation using synthetic plots generated from 1-ha plot data. Notably, the relative error diminishes from approximately 22% to about 5% as plot size increases from 0.04 ha (20x20m) to 0.49 ha (70x70m) across all sites, and we recommend field plots of size ≥0.5 ha for AGB mapping over Tropical Indian Forests. By making this data publicly available, this study paves the way for more accurate spatial AGB mapping in tropical Indian forests, ultimately contributing to better forest management.

Height-diameter models for King Boris fir (Abies borisii regis Mattf.) and Scots pine (Pinus sylvestris L.) in Olympus and Pieria Mountains, Greece

In forest science and practice, the total tree height is one of the basic morphometric attributes at the tree level and it has been closely linked with important stand attributes. In the current research, sixteen nonlinear functions for height prediction were tested in terms of their fitting ability against samples of Abies borisii regis and Pinus sylvestris trees from mountainous forests in central Greece. The fitting procedure was based on generalized nonlinear weighted regression. At the final stage, a five-quantile nonlinear height-diameter model was developed for both species through a quantile regression approach, to estimate the entire conditional distribution of tree height, enabling the evaluation of the diameter impact at various quantiles and providing a comprehensive understanding of the proposed relationship across the distribution. The results clearly showed that employing the diameter as the sole independent variable, the 3-parameter Hossfeld function and the 2-parameter Näslund function managed to explain approximately 84.0% and 81.7% of the total height variance in the case of King Boris fir and Scots pine species, respectively. Furthermore, the models exhibited low levels of error in both cases (2.310m for the fir and 3.004m for the pine), yielding unbiased predictions for both fir (−0.002m) and pine (−0.004m). Notably, all the required assumptions for homogeneity and normality of the associated residuals were achieved through the weighting procedure, while the quantile regression approach provided additional insights into the height-diameter allometry of the specific species. The proposed models can turn into valuable tools for operational forest management planning, particularly for wood production and conservation of mountainous forest ecosystems.

Height-diameter model of broad-leaved mixed forest based on species classification in Maoershan, Northeast China.

The complex stand structure and high species diversity of natural forests pose great challenges for analyzing stand growth and formulating reasonable plans for forest management. The height-diameter relationship is of great significance for predicting stand growth and formulating forest management measures. Based on survey data of 48 broad-leaved mixed forest plots in Maoershan, we classified 23 tree species into four groups based on species structure, growth characteristics and bionomics. We established a generalized model including stand, tree competition, species mixing and species diversity variables by reparameterization method, and a two-level mixed effect model of plot and tree species group. We tested the prediction ability of the model by leave-one-out cross-validation method. The results showed that the Ratkowsky (1990) model was the optimal basic model. The introduction of dominant height, basal area of trees larger than the object tree, basal area proportion of each species, and Shannon index could better explain the height-diameter relationship of broad-leaved mixed forest in Maoershan. The introduction of the mixed effect model of plot and tree species group could significantly improve the prediction accuracy of the model, with a Ra2 of 0.83. Under the same gradient of environmental factors, intolerant tree species exhibited higher tree heights than shade-tolerant tree species. In this study, we used the constructed tree height-diameter model to analyze the effects of species mixing and tree functional traits on tree height, which provided a theoretical basis for accurately predicting height of different tree species and analyzing the growth relationships in broadleaved mixed forests.

Analysis of the inflection points of height-diameter models

The inflection point is an important feature of sigmoidal height-diameter (H-D) models. It is often cited as one of the properties favoring sigmoidal model forms. However, there are very few studies analyzing the inflection points of H-D models. The goals of this study were to theoretically and empirically examine the behaviors of inflection points of six common H-D models with a regional dataset. The six models were the Wykoff (WYK), Schumacher (SCH), Curtis (CUR), Hossfeld IV (HOS), von Bertalanffy-Richards (VBR), and Gompertz (GPZ) models. The models were first fitted in their base forms with tree species as random effects and were then expanded to include functional traits and spatial distribution. The distributions of the estimated inflection points were similar between the two-parameter models WYK, SCH, and CUR, but were different between the three-parameter models HOS, VBR, and GPZ. GPZ produced some of the largest inflection points. HOS and VBR produced concave H-D curves without inflection points for 12.7% and 39.7% of the tree species. Evergreen species or decreasing shade tolerance resulted in larger inflection points. The trends in the estimated inflection points of HOS and VBR were entirely opposite across the landscape. Furthermore, HOS could produce concave H-D curves for portions of the landscape. Based on the studied behaviors, the choice between two-parameter models may not matter. We recommend comparing several three-parameter model forms for consistency in estimated inflection points before deciding on one. Believing sigmoidal models to have inflection points does not necessarily mean that they will produce fitted curves with one. Our study highlights the need to integrate analysis of inflection points into modeling H-D relationships.

Modeling Height–Diameter Allometric Relationships of Dominant African Oak (Afzelia africana Sm) From West African Dry Savannas

Tree height is an important ecological parameter for assessing biomass carbon accumulation rate and understanding the ontogenetic development of vascular plants. Given the lack of height measurement data, height–diameter allometric models are needed to improve carbon stock estimates. In this study, we investigated height–diameter relationships of dominant Afzelia africana Sm in West Africa. Using field measurements of 1483 trees across four forest sites and two climatic zones in Burkina Faso, we (i) compared the performance of 10 theoretical models in predicting height–diameter relationships, (ii) tested for site effects on tree height–diameter allometric relationships, and (iii) evaluated the performance of existing regional height–diameter equation in predicting tree height. We found that the asymptotic Gompertz model better described height–diameter allometric relationships of the studied species in the study area. A significant variation of the Lorey’s mean height and the asymptotic maximum height was observed between forest sites, highlighting the variations of height–diameter allometric relationships with site conditions. Existing height–diameter equation developed for West Africa significantly underestimates tree height of A. africana at between 53.34% and 62.63%. The height–diameter allometric models developed in this study can be applied for carbon stock estimates in the semiarid zones of West Africa. Mixed‐species height–diameter models integrating other sites and varying climatic conditions can improve biomass estimates and contribute to a better understanding of the drivers of height–diameter relationships in savannas.

A generalized mixed-effects height-diameter model for intensively managed Pinus taeda stands in Southern Brazil

A generalized mixed-effects height-diameter model for intensively managed Pinus taeda stands in Southern Brazil

Examining the transferability of height–diameter model calibration strategies across studies

Abstract Height–diameter (H–D) models are fundamental tools for predicting the relationship between tree H–D at breast height, for numerous applications in forestry. Increasingly, studies develop H–D models that can be calibrated to achieve a high level of precision with only a few observations. Different calibration methods and strategies are employed and compared in these studies, often disregarding the data used to develop the models and the H–D function used. In this study, we examined the transferability of optimal calibration strategies across studies, conducting a literature review and an empirical study. We compared the performance of six H–D functions and different calibration methods when using the same calibration strategies and dataset. Based on our literature review, we found that the most commonly employed calibration strategy is random-effects calibration and that the most common variable used to develop generalized H–D models is dominant height. We observed that different calibration methods can lead to varying results due to their different emphases on various aspects of the data and their individual limitations. Moreover, when the same dataset is used for calibration, different H–D functions may exhibit various performances. However, we found high percentages of agreement for the Curtis, Schumacher, and Wykoff H–D functions across all three calibration methods and low agreement between all functions and the Power H–D function. These observations underscore the need to consider all relevant factors, including the H–D function used, when selecting an H–D function and calibration strategy to ensure optimal transferability of the model. Our study provides insights that can improve the accuracy of H–D models, which are essential for predicting forest growth and structure in the context of changing environmental conditions.

Plant functional traits and tree size inequality improved individual tree height prediction of mid-montane humid evergreen broad-leaved forests in southwest China

Tree height and diameter at breast height (DBH) are key survey factors in forest inventory. Compared to measuring DBH, the process of measuring tree height is time-consuming, labor-intensive and costly, especially in natural mixed forests. Tree height is therefore commonly predicted using height-diameter (H-D) models. Nevertheless, developing H-D models for natural mixed forests are hampered by the large diversity of tree species and the small number of rare tree species observed. In this study, a nonlinear mixed effects H-D model was established for a mid-montane humid evergreen broad-leaved forest in southwest China. For this purpose, we classified tree species by K-means clustering according to the specific plant functional traits. Data were collected from 5, 737 trees in 100 subplots with 86 species. The results revealed that the nonlinear mixed effects model of tree height accounted for 75% of the variation in tree height without significant heteroscedasticity, and that the model parameters had biological significance. In the 10-fold cross validation, no clear fluctuation in the model evaluation metrics was observed, indicating the absence of overfitting in the model. Of the 15 stand variables, dominant height (Hd) and Margalef’s index of DBH (DMg), which contributed the most to the height-diameter model, were selected to expand the model. Our findings suggest that tree height increased with increasing values of Hd and DMg, and vice versa. This study highlights the feasibility of categorizing natural mixed forests by utilizing plant functional traits and the influential role of tree size inequality on tree height variation.

Predicting height-diameter relationship in uneven-aged stands in Finland

Rotation forestry has been the main management practice in Finland for the last 50 years, and the modelling of forest growth and yield is mainly based on even-aged stands. Increasing demands for wider application of continuous cover forestry result in the need for models that are applicable to uneven-aged stands. The existing Finnish height-diameter models showed the need for new models or relatively intensive calibration of the existing Näslund's model. Four alternative height curves were tested, namely Näslund's, Prodan's, Hossfeld's and Richards' functions. However, new modifications were implemented for realistic behaviour with small trees. Prediction models were formulated to include stand basal area, basal area-weighted mean dbh and Lorey's height. Including the temperature sum to either intercept or asymptote affected the level of the curve but not its shape. The evaluation of the models showed a good fit with uneven-aged modelling data (ERIKA), but less accurate performance in the test data sets of recently established uneven-aged stand plots on mineral soils or on peatlands. On peatland stands, the existing Näslund's model for even-aged stands performed better than any of the new height-dbh model for uneven-aged stands. This is probably because the growth of smaller trees has recovered better than that of older trees after drainage.

UAV-based height measurement and height–diameter model integrating taxonomic effects: exploring vertical structure of aboveground biomass and species diversity in a Malaysian tropical forest

The complex stratification of tropical forests is a key feature that directly contributes to high aboveground biomass (AGB) and species diversity. This study aimed to explore the vertical patterns of AGB and tree species diversity in the tropical forest of Pasoh Forest Reserve, Malaysia. To achieve this goal, we used a combination of field surveys and drone technology to gather data on species diversity, tree height (H), and tree diameter at breast height (D). As all trees in the 6 ha plot were tagged and identified, we used the data to classify the taxonomy and calculate species diversity indices. We used unmanned aerial vehicle-based structure-from-motion photogrammetry to develop a Digital Canopy Height Model to accurately estimate H. The collected data and previous datasets were then used to develop Bayesian height–diameter (HD) models that incorporate taxonomic effects into conventional allometric and statistical models. The best models were selected based on their performance in cross-validation and then used to estimate AGB per tree and the total AGB in the plot. Results showed that taxonomic effects at the family and genus level improved the HD models and consequent AGB estimates. The AGB was the highest in the higher layers of the forest, and AGB was largely contributed by larger trees, especially specific families such as Dipterocarpaceae, Euphorbiaceae, and Fabaceae. In contrast, species diversity was the highest in the lower layers, whereas functional diversity was the highest in the middle layers. These contrasting patterns of AGB and species diversity indicate different roles of forest stratification and layer-specific mechanisms in maintaining species diversity. This study highlights the importance of considering taxonomic effects when estimating AGB and species diversity in tropical forests. These findings underscore the need for a more comprehensive understanding of the complex stratification of tropical forests and its impact on the forest ecosystem.

Height-diameter models of regenerated saplings of Larix gmelinii based on dummy variable and quantile regression.

Based on data collected from 2054 saplings of Larix gmelinii forest in 55 fixed plots in 2018-2019 in Cuigang Forestry Station, Daxing'anling area, we classified the stand density index (SDI) into four classes, i.e., Class Ⅰ (SDI1<1863 plants·hm-2), Class Ⅱ (1863 plants·hm-2≤SDI2<2155 plants·hm-2), Class Ⅲ (2155 plants·hm-2≤SDI3<2459 plants·hm-2) and Class Ⅳ (SDI4≥2459 plants·hm-2) by using the quartile method. We constructed a dummy variable model and quantile regression model for the height-breast diameter of saplings of L. gmelinii with dummy variable method introduced SDI. The results showed that among the five selected representative non-linear tree height curve models, the Richards model fitted the best, with Ra2, RMSE and MAE of 0.7637, 0.8250 m and 0.5696 m. The dummy variable model including the SDI constructed based on the Richards model showed a 1.3% increase in Ra2 compared with the base model, while RMSE, MAE, and AIC decreased by 2.1%, 1.5%, and 11.2%, respectively. When the quantile τ was 0.5, Ra2 of quantile regression model was the maximum, and RMSE, MAE, AIC was the minimum, being 0.7612, 0.8294 m, 0.5657 m, and -767.19, respectively. Compared with SDI1, sapling height in SDI2-SDI4 was increased by 5.6%, 5.6%, and 11.3%, suggesting reasonable that regulation of stand density was conducive to increase the height growth of saplings in regeneration.

SOIL QUALITY AND HEIGHT DIAMETER MODEL ASSESSMENT IN THE MANAGEMENT OF GMELINA ARBOREA PLANTATION IN NASARAWA STATE UNIVERSITY, NIGERIA

Gmelina arborea is a very important timber species as it provides wood for furniture, serves as carbon sink and creates micro-climate with decrease soil temperatures enabling smaller niches to be formed within the forest ecosystem. Also, the nutrients composition of forest soil determines its stand structure and composition, growth rate and dominance, including other silvicultural practices. The study assessed soil quality and height-diameter (H-D) models in the management of Gmelina arborea plantation in Nasarawa State University. Thirty (30) temporary sample plots of 0.01 ha size were randomly selected from the sampling frame with 30% sampling intensity. Thereafter, the total height and the Dbh of each tree within the selected plots was measured for height-diameter model assessment. Also, ten (10) soil samples were randomly collected from the thirty (30) selected plots for H-D modeling in order to assess the soil properties of the plantation. Soil samples obtained were analyzed in the Faculty of Agriculture Laboratory to obtain data and further subjected to statistical analysis. The results of height-diameter models revealed that model one (1) with Akaike Information Criterion (62.19), Bayesian Information Criterion (66.39) and Residual Standard Error (0.64) had the lowest model selection indices when compared with other four models applied. Therefore, model one (1) was selected as the best and grand model for Gmelina aborea plantation in the study area. Also, the result of soil quality showed a significant difference between the soil properties in the study area. The Least Significant Difference result also showed that...

Incorporated neighborhood and environmental effects to model individual-tree height using random forest regression

ABSTRACT In forest resource inventory, tree height is often estimated by easily measurable diameter from height-diameter model. In this study, we tried to use random forest (RF), an important machine learning method, to model individual-tree height. Results showed that the optimized RF model had better fitting and prediction accuracy (R 2 = 0.8146 and RMSE = 2.2527 m). In terms of relative importance, diameter at breast height (DBH) was the most important factor, followed by neighborhood-related variables and other variables related to environmental conditions. Further, tree height was generally positively affected by DBH, mean diameter of neighbors, DBH dominance, number of neighbors, and mean annual precipitation, but negatively affected by elevation. The results indicated that the RF-based height model was statistically reliable and highly accurate, and it had strong interpretability with ecological significance. Our study will provide a new perspective for the application of machine learning algorithms to forest dynamic modeling.

Determination of optimal tree height models and calibration designs for Araucaria araucana and Nothofagus pumilio in mixed stands affected to different levels by anthropogenic disturbance in South-Central Chile

Key messageHere, we present a workflow for determining the optimal tree height model and calibration design for forests affected to varying degrees by anthropogenic disturbance. For mixed Araucaria-Nothofagus forests, tree height predictions in newly surveyed stands are most accurate and effective when the height of up to five random trees is measured to recalibrate predefined nonlinear mixed-effects models.ContextAraucaria-Nothofagus forests in Chile are affected by anthropogenic disturbances such as intentional forest fires, grazing, and seed harvesting, causing forest structure to become more heterogeneous. This also challenges tree height predictions, which are required for yield estimations, carbon accounting, and forest management, since height measurements of standing trees are often considered too costly, difficult, and imprecise.AimsHow does the structure of these forests vary by different levels of anthropogenic disturbance? Which models for estimating tree height of Araucaria araucana and Nothofagus pumilio are most reliable and generally usable? And considering their application in stands they have not been fitted to, which calibration design is optimal for these models?MethodsTwelve stands were surveyed and classified into four different intensities of anthropogenic disturbance. In 25 to 36 plots per stand, horizontal point sampling measurements of stem diameter as well as of height of selected trees were carried out. Different quantitative stand-level properties were calculated to determine forest structure, which was compared among stands by cluster analysis. To identify the optimal height-diameter (H–D) model, simple models including diameter only as well as generalized models including stand variables were tested, each additionally extended by a nonlinear mixed-effects (NLME) modeling framework accounting for nested and random effects. To further determine tree height in new stands, the optimal model calibration design was identified involving the empirical best unbiased predictor technique.ResultsForest structure greatly varied among stands affected by different levels of anthropogenic disturbance, which challenged the development of tree height prediction models. Of all the simple H–D models considered, the Gompertz model was the best for A. araucana and the Näslund model for N. pumilio. The models progressively improved by adding stand variables and using NLME techniques. However, our final model comparisons indicate that a calibrated simple NLME model without stand variables should be preferred. It was further found that the optimal calibration design is to use five randomly selected trees.ConclusionAlthough anthropogenic disturbances can have a complex effect on height-diameter relationships, the same H–D model can be used for stands representing different anthropogenic disturbance levels and recalibrated by cost-effective measurements.

A local height-diameter model with mixed-effects for Abies religiosa in Tlaxcala, Mexico

A local height-diameter model with mixed-effects for Abies religiosa in Tlaxcala, Mexico