A balance index for phylogenetic trees based on rooted quartets

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 α-model and Aldous' β-model and on arbitrary trees under the α-γ-model. of isomorphisms of the restriction of T to them (the rooted quartet they define), and then we add up these values over all 4-tuples of different leaves of T . The idea behind the definition of this balance index is that a highly symmetrical evolutive process should give rise to symmetrical evolutive histories of many small subsets of taxa. In terms of phylogenetic trees, this leads us to expect that, the most symmetrical a phylogenetic tree is, the most symmetrical will be its restrictions to subsets of leaves of a fixed cardinality. Since the smallest number of leaves yielding enough different tree topologies to allow a meaningful comparison of their symmetry is 4, we assess the balance of a tree by measuring the symmetry of all its rooted quartets and adding up the results. And indeed, in Section 4 below we shall find the trees with maximum and minimum values of our rooted quartet index in both the arbitrary and the bifurcating cases, and it will turn out that the minimum value is reached exactly at the combs (see Fig. 1.(a)), which are usually considered the least balanced trees, and the maximum value is reached, in the arbitrary case, exactly at the rooted stars (see Fig. 1.(b)) and, in the bifurcating case, exactly at the maximally balanced trees (cf. Fig. 3 ), which in both cases are considered the most balanced trees.Besides taking its maximum and minimum values at the expected trees, other important features of our index are that it can be easily computed in linear time and that its mean value and variance can be explicitly computed on any probabilistic model of phylogenetic trees satisfying two natural conditions: independence under relabelings and sampling consistency. This allows us to provide these values for two well-known probabilistic models of bifurcating phylogenetic trees, Ford's α-model [13] and Aldous' β-model [2], which include as specific instances the Yule [14,29] and the uniform [6,24,19] models, as well as for Chen-Ford-Winkel's α-γ-model of multifurcating trees [7]. To our knowledge, this is the first shape index for which closed formulas for the expected value and the variance under the α-γ-model have been provided.The rest of this paper is organized as follows. In the next section we introduce the basic notations and facts on phylogenetic trees that will be used in the rest of the paper, and we recall several preliminary results on probabilistic models of phylogenetic trees, proving those results for which we have not been able to find a suitable reference in the literature. Then, in Section 3, we ...

On Sackin's original proposal: The variance of the leaves' depths as a phylogenetic balance index

“…And for each number n ⩽ 183 of leaves, the minimum V value among all rooted bifurcating trees with n leaves is reached at the maximally balanced trees, those bifurcating trees where the descendant leaves of every internal node split among its pair of children into two subsets of cardinalities differing at most in 1. This is in agreement with other balance indices, like the Sackin index [13], the Colless index [8], the total cophenetic index [27], and the rooted quartets index [9], that classify as most balanced these maximally balanced trees (tied with other trees in the case of Sackin's and Colless' indices, and only them in the case of the other two indices). These maximally balanced trees were called "the most balanced trees" by Shao and Sokal [37].…”

“…These maximally balanced trees were called "the most balanced trees" by Shao and Sokal [37], and they yield the minimum values -among the bifurcating trees with their number of leaves-of the Sackin index [13], the Colless index [8], the total cophenetic index [27], and the rooted quartets index [9]; for the first two balance indices, this minimum value may also be reached at other trees, while for the last two indices it is achieved only at the maximally balanced trees. Moreover, as we mentioned, when n is a power of 2, B n is fully symmetric.…”

“…The degree of balance of a phylogenetic tree is usually measured by means of balance indices. Several such indices have been proposed so far [7,9,14,24,25,27,28,35,37] (see also the section "Measures of overall asymmetry" in [12,Chap. 33]) and Shao and Sokal [37, p. 273] explicitly advised to "choose more than one index" to quantify the balance of a tree.…”

“…To do that, we study what trees achieve the maximum and the minimum values of V for any given number of leaves n: they would play the role of the least and the most balanced rooted trees with n leaves according to V . As far as the maximum value of V goes, it is achieved at the combs, which are the trees classified as most imbalanced by other balance indices like Sackin's [13], Colless' [28], the total cophenetic index [27], the number of cherries (the comb is the only bifurcating tree with a single cherry), or the rooted quartet index [9]. As far as its minimum value goes, in the case of multifurcating trees it is achieved, among other trees, at the rooted star trees (cf.…”

On Sackin's original proposal: The variance of the leaves' depths as a phylogenetic balance index

“…The degree of balance of a phylogenetic tree is usually measured by means of balance indices. Several such indices have been proposed so far [7,9,14,24,25,27,28,35,37] (see also the section "Measures of overall asymmetry" in [12,Chap. 33]) and Shao and Sokal [37, p. 273] explicitly advised to "choose more than one index" to quantify the balance of a tree.…”