A balance index for phylogenetic trees based on rooted quartets

TL;DR

This paper introduces a new balance index for rooted phylogenetic trees based on rooted quartets, applicable to multifurcating trees, with efficient computation and theoretical analysis of its extremal and expected values under various models.

Contribution

It proposes a novel balance index for phylogenetic trees based on rooted quartets, including formulas for its extremal and expected values under multiple evolutionary models.

Findings

Index is computable in linear time.

Maximum and minimum index values are determined.

Expected value and variance formulas are provided for different models.

Abstract

We define a new balance index for rooted phylogenetic trees based on the symmetry of the evolutive history of every set of 4 leaves. This index makes sense for multifurcating trees and it can be computed in time linear in the number of leaves. We determine its maximum and minimum values for arbitrary and bifurcating trees, and we provide exact formulas for its expected value and variance on bifurcating trees under Ford's $\alpha$-model and Aldous' $\beta$-model and on arbitrary trees under the $\alpha$-$\gamma$-model.