Power-laws in phylogenetic trees and the preferential coalescent.

Abstract

Phylogenetic trees capture evolutionary relationships among species and reflect the forces that shaped them. While many studies rely on branch length information, the topology of phylogenetic trees--particularly their degree of imbalance--offers a robust framework for inferring evolutionary dynamics when timing data are uncertain. Classical metrics, such as the Colless and Sackin indices, quantify tree imbalance and have been extensively used to characterize phylogenies. Empirical phylogenies typically show intermediate imbalance, falling between perfectly balanced and highly skewed trees. This regime is marked by a power-law relationship between subtree sizes and their cumulative sizes, governed by a characteristic exponent. Although a recent niche-size model replicates this scaling, its mathematical origin and the exponent's value remain unclear. We present a generative model inspired by Kingman's coalescent that incorporates nichelike dynamics through preferential node coalescence. This process maps to Smoluchowski's coagulation kinetics and is described by a generalized Smoluchowski equation. Our model produces imbalanced trees with power-law exponents matching empirical and numerical observations, revealing the mathematical basis of observed scaling laws and offering new tools to interpret tree imbalance in evolutionary contexts.