Growth equations and curves for citrus trees

Percival Allen, J. H. Gaddum, and S. C Pearce writing in Nature in 1945, all have emphasized the advantages of using the simple and powerful methods afforded by logarithmic transformations in analyzing nonnormal distributions, although it had been amply demonstrated in the relative growth of animal parts by (Huxley (1932)). We have undertaken to illustrate graphically the use of logarithm and power transformations for growth models of trees in orchards and tree organs. Various parameters based on literature, either age or size dependent, are described by power functions, log-log linear curves of the type y = bxk, or semi-log linear curves, exponential functions where y = aebx. Tree height or trunk diameter versus tree age, tree-leaf surface area or number on the tree versus tree age, leaf area versus length, or leaf area versus width are linear log-log functions. It is shown that the first pair of parameters are not normally distributed; latter pairs were demonstrated to be normal. Fruit yield was a nonlinear logarithmic function of tree age and their annual size-frequency distributions were not normal, except for infinitely large populations. Individual fruit size is a linear function of log fruit age, but only the log-log relations are linear for fruit dimensions (diameter, volume, and mass) versus packing number. Log branch fresh weight, leaf and fruit fresh weight are linear functions of log tree age, as are logs of branch dry weight, of branch diameter, of number of branches and surface areas and of volumes. High positive correlations between woody organ ages and dry organ densities invalidate the Rashevsky theoretical growth equation. Insertion of new density terms satisfy validity requirements. Frequency distribution of branch-diameter per tree is a linear log-log function. It is postulated that the linearity of the log-log and semi-log dimensional relationships in plant growth result from similar physical and chemical relationships underlying growth as outlined by kinetic theory.