Revisiting Shao and Sokal's $B_2$ index of phylogenetic balance

TL;DR

This paper reevaluates the $B_2$ index, a probabilistic measure of phylogenetic balance applicable to both trees and networks, analyzing its mathematical properties and biological relevance to promote its renewed use.

Contribution

It provides a comprehensive mathematical analysis of the $B_2$ index and compares its biological relevance to established indices like Colless and Sackin.

Findings

$B_2$ has well-defined expectation and variance under common models.

$B_2$ shows comparable biological relevance to Colless and Sackin indices.

The study advocates for reconsidering the $B_2$ index in phylogenetics.

Abstract

Measures of phylogenetic balance, such as the Colless and Sackin indices, play an important role in phylogenetics. Unfortunately, these indices are specifically designed for phylogenetic trees, and do not extend naturally to phylogenetic networks (which are increasingly used to describe reticulate evolution). This led us to consider a lesser-known balance index, whose definition is based on a probabilistic interpretation that is equally applicable to trees and to networks. This index, known as the $B_2$ index, was first proposed by Shao and Sokal in 1990. Surprisingly, it does not seem to have been studied mathematically since. Likewise, it is used only sporadically in the biological literature, where it tends to be viewed as arcane. In this paper, we study mathematical properties of $B_2$ such as its expectation and variance under the most common models of random trees and its extremal values over various classes of phylogenetic networks. We also assess its relevance in biological applications, and find it to be comparable to that of the Colless and Sackin indices. Altogether, our results call for a reevaluation of the status of this somewhat forgotten measure of phylogenetic balance.