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

Background: The Sackin indexS of a rooted phylogenetic tree, defined as the sum of its leaves' depths, is one of the most popular balance indices in phylogenetics, and Sackin's paper (Syst Zool 21:225-6, 1972) is usually cited as the source for this index. However, what Sackin actually proposed in his paper as a measure of the imbalance of a rooted tree was not the sum of its leaves' depths, but their "variation". This proposal was later implemented as the variance of the leaves' depths by Kirkpatrick and Slatkin in (Evolution 47:1171-81, 1993), where they also posed the problem of finding a closed formula for its expected value under the Yule model.

Sound Colless-like balance indices for multifurcating trees.

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 [Formula: see text] that generalize the Colless index to multifurcating phylogenetic trees. Each [Formula: see text] is determined by the choice of a dissimilarity D and a weight function [Formula: see text].

The expected value of the squared cophenetic metric under the Yule and the uniform models.

The cophenetic metrics d, for p ∈ {0} ∪ [1, ∞), are a recent addition to the kit of available distances for the comparison of phylogenetic trees. Based on a fifty years old idea of Sokal and Rohlf, these metrics compare phylogenetic trees on a same set of taxa by encoding them by means of their vectors of cophenetic values of pairs of taxa and depths of single taxa, and then computing the L norm of the difference of the corresponding vectors. In this paper we compute the expected value of the square of d on the space of fully resolved rooted phylogenetic trees with n leaves, under the Yule and the uniform probability distributions.

Cophenetic metrics for phylogenetic trees, after Sokal and Rohlf.

Background: Phylogenetic tree comparison metrics are an important tool in the study of evolution, and hence the definition of such metrics is an interesting problem in phylogenetics. In a paper in Taxon fifty years ago, Sokal and Rohlf proposed to measure quantitatively the difference between a pair of phylogenetic trees by first encoding them by means of their half-matrices of cophenetic values, and then comparing these matrices. This idea has been used several times since then to define dissimilarity measures between phylogenetic trees but, to our knowledge, no proper metric on weighted phylogenetic trees with nested taxa based on this idea has been formally defined and studied yet.