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 that generalize the Colless index to multifurcating phylogenetic trees. Each is determined by the choice of a dissimilarity D and a weight function . 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 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 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.
Highlights
Since the early 1970s, the shapes of phylogenetic trees have been used to test hypothesis about the evolutive forces underlying their assembly [1]
We show that the choice of the dissimilarity D does not cause any major difference in the maximally unbalanced trees relative to CD;f for a fixed f, but that changing the function f implies completely different maximally unbalanced trees
To assess the performance of C, MDM;lnðnþeÞ which we abbreviate by C, we downloaded (December 13-14, 2015) all phylogenetic trees in the TreeBASE database [11] using the function search_treebase() of the R package treebase [22]
Summary
Since the early 1970s, the shapes of phylogenetic trees have been used to test hypothesis about the evolutive forces underlying their assembly [1]. The most used topological feature of phylogenetic trees in this regard is their symmetry, which captures the symmetry of the evolutionary histories described by them. The symmetry of a tree is usually measured through its balance Several balance indices have been proposed so far to quantify the balance of a phylogenetic tree.
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