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& Context Families of the Gumbel (type I), Fréchet (type II) and Weibull (type III) distributions can be combined in the generalized extreme value (GEV) family of distributions. Maximum and minimum values of diameters in forest stands can be used in forest modelling, mainly to define parameters of the functions used in diameter class models as well as in some practical cases, such as modelling maximum diameters for sawing and processing purposes. & Aims The purpose of this study was to examine and compare two extreme value distribution functions (the Gumbel and the Weibull functions) in modelling the distribution of the minimum and the maximum values of representative sets of tree diameter samples. Both of these functions were applied to the lower and upper values of the diameter distributions of the main forest species in northwest Spain: Quercus robur L., Betula pubescens Ehrh., Pinus radiata D. Don, Pinus pinaster Ait. and Pinus sylvestris L. & Methods Parameters of the Gumbel function were estimated using the mode and the moments of the distributions, and parameters of the Weibull function were estimated using the moments method.& Results In general, the Weibull distribution was the most suitable model for describing the maximum diameters. The mode method of the Gumbel yielded the best results for minimum diameters of birch and Monterrey pine. The Gumbel distribution, fitted by either the mode-or momentsbased methods, proved more suitable than the Weibull distribution for describing the minimum diameters in maritime pine and Scots pine stands. & Conclusion In some cases, better results were obtained with the Gumbel than the Weibull distribution for describing the distribution of extreme diameter values in forest stands in northwest Spain. This is the first example of the application of the Gumbel distribution in forest modelling.

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Site index curves were created for natural Aleppo pine (Pinus halepensis Mill.) stands in the Ebro Valley (northeastern Spain). Data were obtained from 54 felled dominant trees. The Generalized Algebraic Difference Approach (GADA) was used to fit 11 equations with a longitudinal data structure that included all possible growth intervals. Three statistical criteria were used for model comparison: root mean square error (RMSE), adjusted coefficient of determination (R2adj) and Akaike’s information criterion (AIC). Graphical evaluation of the data (plots of observed against predicted values and of residuals against predicted values) was also conducted. In addition, the root mean square error (RMSE) was plotted against age. Finally, fitted site index curves for different site qualities were superimposed on the profile plots of the stems analysed. The best reference age (60 years) was calculated from the relative error in the dominant height prediction. The Hossfeld IV model proved the most suitable for representing site index in the study area. Furthermore, only three site index curves (6, 10 and 14 m at 60 years) were suitable for classifying the entire study area because of the low productivity of the stands. These were compared with other site index curves developed for Spain and other countries, for the natural range of distribution of the species. Specific site index curves must be created for the study area, because the existing qualities and growth patterns are not well-represented in other models.

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 (Dn), 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·dmin). 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 dg, dmed and dmax. 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.

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.

Aim of study: In this study we compare the accuracy of three bivariate distributions: Johnson's S BB , Weibull-2 P and LL-2 P functions for characterizing the joint distribution of tree diameters and heights.Area of study: North-West of Spain. Material and methods:Diameter and height measurements of 128 plots of pure and even-aged Tasmanian blue gum (Eucalyptus globulus Labill.) stands located in the North-west of Spain were considered in the present study. The S BB bivariate distribution was obtained from S B marginal distributions using a Normal Copula based on a four-parameter logistic transformation. The Plackett Copula was used to obtain the bivariate models from the Weibull and Logit-logistic univariate marginal distributions. The negative logarithm of the maximum likelihood function was used to compare the results and the Wilcoxon signed-rank test was used to compare the related samples of these logarithms calculated for each sample plot and each distribution.Main results: The best results were obtained by using the Plackett copula and the best marginal distribution was the Logit-logistic. Research highlights:The copulas used in this study have shown a good performance for modeling the joint distribution of tree diameters and heights. They could be easily extended for modelling multivariate distributions involving other tree variables, such as tree volume or biomass.