Simple models for scaling in phylogenetic trees

TL;DR

This paper investigates why certain biological trees, like phylogenetic trees, do not follow the typical logarithmic depth scaling, by analyzing two models that exhibit power-law scaling of tree depth.

Contribution

The paper introduces a new activity model demonstrating power-law depth scaling at a critical parameter, expanding understanding of tree growth dynamics.

Findings

Ford's alpha model approaches asymptotic behavior only at very large sizes

The new activity model exhibits power-law scaling at a critical point

Numerical analysis supports analytical predictions of non-logarithmic scaling

Abstract

Many processes and models --in biological, physical, social, and other contexts-- produce trees whose depth scales logarithmically with the number of leaves. Phylogenetic trees, describing the evolutionary relationships between biological species, are examples of trees for which such scaling is not observed. With this motivation, we analyze numerically two branching models leading to non-logarithmic scaling of the depth with the number of leaves. For Ford's alpha model, although a power-law scaling of the depth with tree size was established analytically, our numerical results illustrate that the asymptotic regime is approached only at very large tree sizes. We introduce here a new model, the activity model, showing analytically and numerically that it also displays a power-law scaling of the depth with tree size at a critical parameter value.