Effect of a heterogeneous site size distribution on the branch size distribution of tree networks

In this study, we investigate how the heterogeneous site size distribution affects the behavior of the branch size distribution of critical trees. By assuming a power-law, P(s) ~ s ^-α, for the site size distribution, we find that the exponent τ for the branch size distribution is smaller than that of a critical tree when α < 3. When α ≥ 3, we find τ = 1. 5, which is the value of the critical tree for P(s) = δ(s - 1). We also find τ = 1. 5 for other P(s)'s with finite 〈s ²〉. Based on our measurement, the physical origin of the discrepancy between the measured branch size distribution in real river networks and that of the critical tree is suggested. As an extension of our study, we also measure the directory size distribution of computer operating systems and discuss the effect of the heterogeneity of the site size distribution τ of a supercritical tree.