Split-Facets for Balanced Minimal Evolution Polytopes and the Permutoassociahedron

Understanding the face structure of the balanced minimal evolution (BME) polytope, especially its top-dimensional facets, is a fundamental problem in phylogenetic theory. We show that BME polytope has a sublattice of its poset of faces which is isomorphic to a quotient of the well-studied permutoassociahedron. This sublattice corresponds to compatible sets of splits displayed by phylogenetic trees and extends the lattice of faces of the BME polytope found by Hodge, Haws and Yoshida. Each of the maximal elements in our new poset of faces corresponds to a single split of the leaves. Nearly all of these turn out to actually be facets of the BME polytope, a collection of facets which grows exponentially.