The minimum value of the Colless index

TL;DR

This paper determines the minimum Colless index value for all rooted bifurcating trees with n leaves, extending previous knowledge limited to cases where n is a power of 2, and characterizes the trees achieving this minimum.

Contribution

It provides a formula for the minimum Colless index for any number of leaves and characterizes the trees that attain this minimum value.

Findings

Derived a formula for the minimum Colless index for all n

Characterized trees that achieve the minimum Colless index

Extended known results beyond powers of 2

Abstract

The Colless index is one of the oldest and most widely used balance indices for rooted bifurcating trees. Despite its popularity, its minimum value on the space $\mathcal{T}_n$ of rooted bifurcating trees with $n$ leaves is only known when $n$ is a power of 2. In this paper we fill this gap in the literature, by providing a formula that computes, for each $n$, the minimum Colless index on $\mathcal{T}_n$, and characterizing those trees where this minimum value is reached.