  Understanding the face structure of the balanced minimal evolution (BME)
polytope, especially its top-dimensional facets, is crucially important to
phylogenetic applications. We show that BME polytope has a sub-lattice of its
poset of faces which is isomorphic to a quotient of the well-studied
permutoassociahedron. This sub-lattice 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.
