A new balance index for phylogenetic trees

• Published in: Mathematical biosciences Vol. 241; no. 1; pp. 125 - 136
• Main Authors: Mir, Arnau, Rosselló, Francesc, Rotger, Lucı´a
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 01.01.2013
• Subjects: Balance index; Biological Evolution; Cophenetic value; Evolution; Mathematical Concepts; Models, Genetic; Phylogenetic tree; Phylogeny; Sackin index; Balance index; Cophenetic value; Sackin index; Phylogenetic tree
• ISSN: 0025-5564; 1879-3134
• DOI: 10.1016/j.mbs.2012.10.005

► We define a new balance index for phylogenetic trees. ► We show that it has greater resolution power than Sackin’s and Colless’ indices. ► We compute its range of values for arbitrary and binary trees. ► We obtain exact formulas for its expected value under the uniform and the Yule models. ► We obtain a formula for the expected value of Sackin’s index under the uniform model. Several indices that measure the degree of balance of a rooted phylogenetic tree have been proposed so far in the literature. In this work we define and study a new index of this kind, which we call the total cophenetic index: the sum, over all pairs of different leaves, of the depth of their lowest common ancestor. This index makes sense for arbitrary trees, can be computed in linear time and it has a larger range of values and a greater resolution power than other indices like Colless’ or Sackin’s. We compute its maximum and minimum values for arbitrary and binary trees, as well as exact formulas for its expected value for binary trees under the Yule and the uniform models of evolution. As a byproduct of this study, we obtain an exact formula for the expected value of the Sackin index under the uniform model, a result that seems to be new in the literature.