Two Results about the Sackin and Colless Indices for Phylogenetic Trees and Their Shapes

TL;DR

This paper provides new asymptotic and exact formulas for the expected Sackin and Colless indices of phylogenetic tree shapes and labeled trees, enhancing understanding of tree balance metrics in evolutionary studies.

Contribution

It offers the first asymptotic analysis for unlabelled tree shapes and a simple proof for the expected Sackin index in labeled phylogenetic trees under the uniform model.

Findings

Asymptotic expected indices for unlabelled tree shapes

Elementary proof of the expected Sackin index formula

Enhanced understanding of tree balance metrics

Abstract

The Sackin and Colless indices are two widely-used metrics for measuring the balance of trees and for testing evolutionary models in phylogenetics. This short paper contributes two results about the Sackin and Colless indices of trees. One result is the asymptotic analysis of the expected Sackin and Colless indices of a tree shape (which are full binary rooted unlabelled trees) under the uniform model where tree shapes are sampled with equal probability. Another is a short elementary proof of the closed formula for the expected Sackin index of phylogenetic trees (which are full binary rooted trees with leaves being labelled with taxa) under the uniform model.