Will forest size structure follow the -2 power-law distribution under ideal demographic equilibrium state?

Scaling relations formed in forest development processes are fairly important for understanding and predicting forest dynamics. During self-thinning of a relatively even-sized forest, tree abundance will decrease with an increase in average tree size, forming the size-abundance relation (SAR); while for a size-structured forest under the demographic equilibrium state, the frequency of trees also varies with size classes in a similar, decreasing pattern, manifesting as the size-frequency distribution (SFD). In the metabolic scaling theory (MST), the two scaling relations are considered to be consistent. However, in this paper, we proved that SFD can never be equivalent to SAR unless the growth rate of tree diameters is a constant. The reason derives from the time differences of transition between different size classes, which are influenced in SFD maintenance but not in SAR formation. Demographic equilibrium of a size structured forest requires a different resource allocation among different size classes at the same time, which contradicts the resource conservation during SAR formation in the self-thinning process. Consequently, if the rate of resource use per individual scales as a +2 power with its diameter according to MST, which led to the -2 power SAR, SFD can never be a -2 power-law distribution. The previous confusion between SFD of a size-structured forest and SAR formed during self-thinning processes may lead to many misunderstandings and unreliable predictions on forest structure and dynamics.