The minimum value of the Colless index

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.