On the minimum value of the Colless index and the bifurcating trees that achieve it

Abstract

Measures of tree balance play an important role in the analysis of phylogenetic trees. One of the oldest and most popular indices in this regard is the Colless index for rooted bifurcating trees, introduced by Colless [8]. While many of its statistical properties under different probabilistic models for phylogenetic trees have already been established, little is known about its minimum value and the trees that achieve it. In this manuscript, we fill this gap in the literature. To begin with, we derive both recursive and closed expressions for the minimum Colless index of a tree with n leaves. Surprisingly, these expressions show a connection between the minimum Colless index and the so-called Blancmange curve, a fractal curve. We then fully characterize the trees that achieve this minimum value and we introduce both an algorithm to generate them and a recurrence to count them. After focusing on two extremal classes of trees with minimum Colless index (the maximally balanced trees and the greedy from the bottom trees), we conclude by showing that all trees with minimum Colless index also have minimum Sackin index, another popular balance index.

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.…”

On the Colijn-Plazzotta numbering scheme for unlabeled binary rooted trees

“…The study augments recent results examining unlabeled binary rooted trees that possess maximal or minimal features in scenarios arising from consideration of evolutionary problems [6,7,8,13]. Curiously, Theorem 10 has a close connection with an analysis of "non-equivalent ancestral configurations," structures that are used in characterizing relationships of pairs of trees [9,16].…”

“…Thus, because the most extreme CP ranks among trees of size n are represented by a balanced tree and the unbalanced caterpillar tree, CP rank has potential to be useful in the measurement of tree balance-the extent to which an unlabeled shape resembles balanced shapes [2,10,12,15]. Because the tree of minimal CP rank has absolute difference 0 or 1 between the sizes of the two subtrees for each internal node, it is perhaps useful to consider CP rank specifically in relation to the Colless tree balance index [3,5,6,14]-which for each node sums the absolute difference in the numbers of descendants of the two subtrees of the node and which has larger values for unbalanced trees.…”

“…The sequence of values U n , the Wedderburn-Etherington numbers, begins from n = 1 with 1, 1, 1, 2, 3, 6,11,23,46,98,207,451,983,2179,4850 (Table 2, A001194 in OEIS). U n is straightforward to calculate from U 1 , U 2 , .…”