The minimum value of the Colless index

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.