Tree height-diameter (H-D) relationships are important for routine forest assessment. Several H-D relationships have been developed for different species and more are still evolving. This study introduces new H-D model developed for Pinus pinaster and Pinus radiata in Spain, based on data from 184 and 96 permanent sample plots, respectively, collected in the northwest region of the country. Nonlinear mixed-effect modelling technique was used to fit the generalized H-D model. The mixed-effect H-D model was calibrated using the random effects predicted from one to three randomly selected trees per sample plot. Different indices including root mean square error (RMSE) and adjusted coefficient of determination () were used to assess the predictive performance of the model. The results showed that the new model had and RMSE of 0.906 and 1.156 m and 0.814 and 1.703 m for P. pinaster and P. radiata, respectively. The calibration response involved the selection of one tree per sample plot and resulted in a reduction of RMSE by 6.5% and 13.5% for pinaster and P. radiata, respectively.

Aim of study: In this study we compared the accuracy of the Weibull and the Johnson’s SB functions for describing diameter distributions in pedunculate oak (Quercus robur L.) and birch (Betula pubescens Ehrh.) stands.Area of study: Galicia (Northwest Spain).Material and Methods: A total of 172 diameter distributions in pedunculate oak and 202 in birch stands were finally evaluated. We compared the accuracy of three commonly used estimation methods of the Weibull and four estimation methods of the Johnson’s SB functions for describing these diameter distributions.Main results. For Quercus robur L. stands, the most suitable methods were the Percentiles followed by Maximum Likelihood for the Weibull PDF and the method of Moments for the Johnson’s SB PDF. For Betula pubescens Ehrh. stands, the best fits obtained with the Percentiles and Maximum Likelihood methods were also superior to the method of Moments, whereas the Conditional Maximum Likelihood and method of Moments provided the best results for the Johnson’s SB PDF, depending on the statistic and the value of the location parameter considered.Research highlights: Both distributions were suitable. The results were better for pedunculate oak than for birch stands.Keywords: Knoebel and Burkhart; location parameter; percentiles; maximum likelihood; moments; mode.

The diameter distributions of trees in 50 temporary sample plots (TSPs) established in Pinus halepensis Mill. stands were recovered from LiDAR metrics by using six probability density functions (PDFs): the Weibull (2P and 3P), Johnson’s SB, beta, generalized beta and gamma-2P functions. The parameters were recovered from the first and the second moments of the distributions (mean and variance, respectively) by using parameter recovery models (PRM). Linear models were used to predict both moments from LiDAR data. In recovering the functions, the location parameters of the distributions were predetermined as the minimum diameter inventoried, and scale parameters were established as the maximum diameters predicted from LiDAR metrics. The Kolmogorov–Smirnov (KS) statistic (D_n), number of acceptances by the KS test, the Cramér von Misses (W²) statistic, bias and mean square error (MSE) were used to evaluate the goodness of fits. The fits for the six recovered functions were compared with the fits to all measured data from 58 TSPs (LiDAR metrics could only be extracted from 50 of the plots). In the fitting phase, the location parameters were fixed at a suitable value determined according to the forestry literature (0.75·d_min). The linear models used to recover the two moments of the distributions and the maximum diameters determined from LiDAR data were accurate, with R² values of 0.750, 0.724 and 0.873 for d_g, d_med and d_max. Reasonable results were obtained with all six recovered functions. The goodness-of-fit statistics indicated that the beta function was the most accurate, followed by the generalized beta function. The Weibull-3P function provided the poorest fits and the Weibull-2P and Johnson’s SB also yielded poor fits to the data.

Modeling diameter distribution of forest stands requires suitable function(s) with the appropriate parameter estimation methods. To date, the parameters of most familiar functions in forestry have been estimated with derivative methods such as moment, percentiles, conditional maximum likelihood, etc. with little emphasis on numerical optimization. Therefore, this study compares the suitability of derivative and numerical methods to estimate diameter distribution of forest stands. The derivative and optimization methods were used to fit six commonly used functions in forestry including Weibull (2 and 3-parameter), Johnson’s SB, beta, generalized beta and gamma. The data comprised 303 and 96 permanent sample plots temperate forest (Eucalyptus globulus and Pinus radiata, respectively) and 65 temporary sample plots from tropical forest (Gmelina arborea). Different indices such as Kolmogorov–Smirnov, Cramer-von Mises and mean squared error were used to assess the methods. The results show that the derivative method by moments provided the best fit for beta and Johnson’s S_B distributions. The optimization methods (“lifereg” and “optim”) were more suitable for the Weibull and gamma distributions. Both methods were appropriate for the generalized beta distribution. Abbreviation: PW2P: Percentile Weibull two-parameter; MW2P: Moments Weibull two-parameter; MLLW2P: Maximum likelihood “lifereg” Weibull two-parameter; MLOW2P: Maximum likelihood “optim” Weibull two-parameter; PW3P: Percentile Weibull three-parameter; MW3P: Moments Weibull three-parameter; MLLW3P: Maximum likelihood “lifereg” Weibull three-parameter; MLOW3P: Maximum likelihood “optim” Weibull three-parameter; CMLJ: Conditional maximum likelihood Johnson’s SB; MJ Moments Johnson’s SB; MLOJ: Maximum likelihood “optim” Johnson’s SB; MB: Moments beta; MLOB Maximum likelihood “optim” beta; MGB: Moments generalized beta; MLOGB Maximum likelihood “optim” generalized beta; MG2P: Moments gamma two-parameter; MLOG2P: Maximum likelihood “optim” gamma two-parameter

Modeling diameter distribution is a crucial aspect of forest management, requiring the selection of an appropriate probability density function or cumulative distribution function along with a fitting method. This study compared the suitability of eight probability density functions—A Charlier, beta, generalized beta, gamma, Gumbel, Johnson’s SB, and Weibull (two- and three-parameter)—fitted using both derivative methods (Moments) fitted in SAS/STAT^TM and optimization methods (MLE) fitted with the ‘optim’ function in R for diameter distribution estimation in forest stands. The A Charlier and Gumbel functions were used for the first time in this type of comparison. The data were derived from 167 permanent sample plots in an Atlantic forest (Quercus robur) and 59 temporary sample plots in tropical forests (Tectona grandis). Fit quality was assessed using various indices, including Kolmogorov–Smirnov, Cramér–von Mises, mean absolute error, bias, and mean squared error. The results indicated that Johnson’s SB function was more suitable for describing the diameter distribution of the stands. Johnson’s SB, three-parameter Weibull, and generalized beta consistently performed well across different fitting methods, while the fits produced by gamma, Gumbel, and two-parameter Weibull were of poor quality.

Aim of study: In this study, both the direct and indirect methods by conditional maximum likelihood (CML) and moments for fitting Johnson’s SBB were evaluated. To date, Johnson’s SBB has been fitted by either indirect (two-stage) method using well-known procedures for the marginal diameter and heights, or direct methods, where all parameters are estimated at once. Application of bivariate Johnson’s SBB for predicting height and improving volume estimation requires a suitable fitting method.Area of study: E. globulus, P. pinaster and P. radiata stands in northwest Spain.Material and methods: The data set comprised of 308, 184 and 96 permanent sample plots (PSPs) from the aforementioned species. The suitability of the method was evaluated based on height and volume prediction. Indices including coefficient of determination (R2), root mean square Error (RMSE), model efficiency (MEF), Bayesian Information Criterion (BIC) and Hannan-Quinn Criterion (HQC) were used to assess the model predictions. Significant difference between observed and predicted tree height and volumes were tested using paired sample t-test at 5% level for each plot by species.Main results: The indirect method by CML was the most suitable method for height and volume prediction in the three species. The R2 and RMSE for height prediction ranged from 0.994 – 0.820 and 1.454 – 1.676, respectively. The percentage of plot in which the observed and predicted heights were significant was 0.32%. The direct method was the least performed method especially for height prediction in E. globulus.Research highlights: The indirect (two-stage) method, especially by conditional maximum likelihood, was the most suitable method for the bivariate Johnson’s SBB distribution.Keywords: conditional maximum likelihood; moments; two-stage method; direct method; tree volume.