Sound Colless-like balance indices for multifurcating trees

Arnau Mir, Lucía Rotger and Francesc Rosselló

PLOS ONE, 2018, vol. 13, issue 9, 1-27

Abstract: The Colless index is one of the most popular and natural balance indices for bifurcating phylogenetic trees, but it makes no sense for multifurcating trees. In this paper we propose a family of Colless-like balance indices C D , f that generalize the Colless index to multifurcating phylogenetic trees. Each C D , f is determined by the choice of a dissimilarity D and a weight function f : N → R ≥ 0. A balance index is sound when the most balanced phylogenetic trees according to it are exactly the fully symmetric ones. Unfortunately, not every Colless-like balance index is sound in this sense. We prove then that taking f(n) = ln(n + e) or f(n) = en as weight functions, the resulting index C D , f is sound for every dissimilarity D. Next, for each one of these two functions f and for three popular dissimilarities D (the variance, the standard deviation, and the mean deviation from the median), we find the most unbalanced phylogenetic trees according to C D , f with any given number n of leaves. The results show that the growth pace of the function f influences the notion of “balance” measured by the indices it defines. Finally, we introduce our R package “CollessLike,” which, among other functionalities, allows the computation of Colless-like indices of trees and their comparison to their distribution under Chen-Ford-Winkel’s α-γ-model for multifurcating phylogenetic trees. As an application, we show that the trees in TreeBASE do not seem to follow either the uniform model for multifurcating trees or the α-γ-model, for any values of α and γ.

Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:plo:pone00:0203401

DOI: 10.1371/journal.pone.0203401

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