Several probability density functions for statistical distributions have been used to model the distribution of tree diameters. By considering transformations of variables which preserve the functional form of the distribution, tree diameter growth models are derived. If X is the tree's diameter at the beginning of a growth period and I is the increment in diameter, then the model I = (β0 - X) + β1(X - β3)β2 is implied by assuming the Weibull, lognormal, or generalized gamma distribution as a diameter distribution model. A special case (β2 = 1) derived by Meyer (1952) applies when the exponential, normal, beta, or Johnson's SB is assumed. Biologically the equation can be shown to state that when β2 = 1 the relative rate of diameter growth is constant over age. With individual tree data from a spacing study of loblolly pine, β2 = 1 was rejected. These results provide a link between tree-level growth, models and stand-level diameter distribution models. Forest Sci. 26:626-632.
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Robert L. Bailey (1980) studied this question.