Abstract. The balanced minimal evolution (BME) method of creating phylogenetic trees can be formulated as a linear programming problem, minimizing an inner product over the vertices of the BME polytope. In this paper we undertake the project of describing the facets of this polytope. We classify and identify the combinatorial structure and geometry (facet inequalities) of all the facets in dimensions up to 5, and classify even more facets in all dimensions. A full set of facet inequalities would allow a full implementation of the simplex method for finding the BME tree-although there are reasons to think this an unreachable goal. However, our results provide the crucial first steps for a more likely-to-be-successful program: finding efficient relaxations of the BME polytope.

Abstract. 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 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.

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