Exact formulas for the variance of several balance indices under the Yule model

TL;DR

This paper derives exact formulas for the variance of several phylogenetic balance indices under the Yule model, enabling more precise statistical tests of evolutionary tree balance regardless of tree size.

Contribution

It provides the first exact formulas for the variance of the Sackin, Colless, and total cophenetic indices under the Yule model for any number of leaves.

Findings

Exact variance formulas for the indices

Covariance between Sackin and cophenetic indices

Applicable to trees of any size

Abstract

One of the main applications of balance indices is in tests of null models of evolutionary processes. The knowledge of an exact formula for a statistic of a balance index, holding for any number n of leaves, is necessary in order to use this statistic in tests of this kind involving trees of any size. In this paper we obtain exact formulas for the variance under the Yule model of the Sackin index, the Colless index and the total cophenetic index of binary rooted phylogenetic trees with n leaves. We also obtain the covariance of the Sackin and the total cophenetic index.