  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
(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.
