ArticleJournal of mathematical biology2020
On the minimum value of the Colless index and the bifurcating trees that achieve it.
Article in Journal of mathematical biology, 2020. The graph could read no effect estimate from its abstract, so it casts no vote on the map. Cited by 7 papers.
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7 citing papers in PubMed.
- Metaconcepts of Rooted Tree Balance.Bulletin of mathematical biology · 2025Article
- Article
- Explicit solution of divide-and-conquer dividing by a half recurrences with polynomial independent term.PloS one · 2022Article
- A simple derivation of the mean of the Sackin index of tree balance under the uniform model on rooted binary labeled trees.Mathematical biosciences · 2021Article
- Revisiting Shao and Sokal'sJournal of mathematical biology · 2021Article
- On the Colijn-Plazzotta numbering scheme for unlabeled binary rooted trees.Discrete applied mathematics (Amsterdam, Netherlands : 1988) · 2021Article
- On Sackin's original proposal: the variance of the leaves' depths as a phylogenetic balance index.BMC bioinformatics · 2020Article
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5 authors.
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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 (Syst Zool 31:100-104, 1982). 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 tree shapes 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.
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