Probability Distributions in High-Density Dendroenergy Plantations Simo ´n Sandoval, Jorge Cancino, Rafael Rubilar, Edwin Esquivel, Eduardo Acun ˜a, Fernando Mun ˜ oz, and Miguel Espinosa Abstract: Six probability density functions were used to model diametric distributions of Acacia melanoxylon, Eucalyptus camaldulensis, and Eucalyptus nitens being investigated for dendroenergy purposes at three plan- tation densities (5,000, 7,500, and 10,000 trees ha 1 ). The Weibull was the function with the best fit, followed in decreasing order by the beta, Johnson S B , gamma, lognormal, and Johnson S U functions. Planting density affected the shape and amplitude of the diametric distribution of all species. Increasing stocking made distributions more leptokurtic and narrower. Analyses over time during the first 28 months suggested a stronger effect on diameter distribution form, which was less evident during the early ages of the crop. FOR.SCI. 58(6): 663– 672. Keywords: probability density functions, short rotation crops, bioenergy, stocking, Weibull D ECISIONMAKING IN FORESTRY is often based largely on a plantation’s growth and yield (Parresol 2003), and the prediction of these variables has provided a constant focus for studies. At present, mathe- matical modeling is used to predict the growth and yield of a given plantation, considering stand variables such as stocking, basal area, dominant height, and diametric fre- quency distribution per unit area (Gove and Patil 1998). Knowing the diametric distribution of a plantation, as well as the distribution of other variables at the tree level (e.g., volume and biomass), constitutes a fundamental tool for decisionmaking in forest management (Zhang et al. 2003, Cao 2004) and is one of the main characteristics used to determine stand state variables such as basal area, volume, or biomass per unit area (Mehta ¨talo 2004). Different theoretical probability density functions (PDFs) have been used to describe the diametric distribution of plantations, including the beta (Clutter and Bennett 1965, Lenhart and Clutter 1971, Zohrer 1972, Li et al. 2002), gamma (Nelson 1964), lognormal (Bliss and Reinker 1964), Weibull (Bailey and Dell 1973, Rennolls et al. 1985), and Johnson (Hafley and Schreuder 1977, Zhou and McTague 1996, Kamziah et al. 1999). Clutter and Bennett (1965) were pioneers introducing the diametric distribution meth- odology in growth and yield models. These authors used the four-parameter beta PDF to describe the distribution of the number of trees per unit area for classes of diameter at breast height (dbh). Since then, PDFs have been widely used to model plantation growth and yield (Lindsay et al. 1996). Bailey and Dell (1973) were the first to use the Weibull PDF as a dbh frequency distribution model, noting that this function had certain advantages over the beta function; because the Weibull PDF has a closed form, it can be expressed as a cumulative density function that can be evaluated without numerical integration and only requires the estimation of three parameters. Hafley and Schreuder (1977) introduced the Johnson S B distribution (system bounded) (Johnson 1949) to the methodology of diametric distributions. Alzaid and Sultan (2009) analyzed the gamma and lognormal PDFs and recommended the use of the gamma distribution, because of its higher flexibility. The analysis and selection of the right PDF for a data set has been widely discussed in different research areas. Sev- eral authors (Cox 1961, 1962, Chambers and Cox 1967, Atkinson 1969, 1970, Dyer 1973, Chen 1980) highlighted the importance of selecting the optimum PDF because of the better utility of these functions. Moreover, several studies have focused on describing differences among PDFs (Jack- son 1969, Dumonceaux and Antle 1973, Bain and Engel- hard 1980, Fearn and Nebenzahl 1991, Wiens 1999, Gupta and Kundu 2003a, 2003b, 2004, Alzaid and Sultan 2009). Although numerous research cases in forestry address this topic, they focus nearly exclusively on using PDFs to de- scribe the diametric distributions of traditional plantations (i.e., stands for sawtimber or pulp) and on modeling the diametric distribution at a specific site without comparing PDFs in terms of fit quality and precision (Hafley and Schreuder 1977, Reynolds 1984, Newberry and Burk 1985, Reynolds et al. 1988, Lindsay et al. 1996, Zhang et al. 2003, Cao 2004, Lei 2008). Studies published on this area have not considered plantations for biomass production for den- droenergy grown at high stockings for short rotation har- vesting periods. In Chile, crops for biomass production for dendroenergy Manuscript received March 7, 2011; accepted March 21, 2012; published online April 26, 2012; http://dx.doi.org/10.5849/forsci.11-028. Simo ´n Sandoval, Universidad de Concepcio ´n, Victoria 631, Barrio Universitario, Concepcio ´n, Concepcio ´n, Chile—Phone: 2204979; sisandoval@udec.cl. Jorge Cancino, Universidad de Concepcio ´ n—jcancino@udec.cl. Rafael Rubilar, Universidad de Concepcio ´ n—rrubilar@ncsfnc.cfr.ncsu.edu. Edwin Esquivel, Universidad de Concepcio ´n— hddedwin@gmail.com. Eduardo Acun ˜a, Universidad de Concepcio ´n— edacuna@udec.cl. Fernando Mun ˜oz, Universidad de Concepcio ´n—fmunoz@udec.cl. Miguel Espinosa, Universidad de Concepcio ´n—mespinos@udec.cl. Acknowledgments: This work was supported by the project INNOVA Bio-Bio N o 06 PC S1-33. This article uses metric units; the applicable conversion factors are: millimeters (mm): 1 mm  0.039 in.; centimeters (cm): 1 cm  0.39 in.; meters (m): 1m  3.3 ft.; hectares (ha): 1 ha  2.47 ac.; kilograms (kg): 1 kg  2.2 lb. Copyright © 2012 by the Society of American Foresters. Forest Science 58(6) 2012 663