The Bandelt-Dress Quartet Distance Conjecture

The quartet distance counts the four-leaf subsets on which two binary phylogenetic trees display different topologies. Bandelt and Dress conjectured the maximum over trees on $n$ leaves. Proved: it is $(2/3 + o(1))\binom{n}{4}$, by reducing arbitrary pairs of trees to caterpillars through a common-root planarization and an identity on five-leaf trees.