The Bergman complex of a matroid and phylogenetic trees

We study the Bergman complex B(M) of a matroid M: a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of the proper part of its lattice of flats. In addition, we show that the Bergman fan of the graphical matroid of the complete graph Kn is homeomorphic to the space of phylogenetic trees Tn×R. This leads to a proof that the link of the origin in Tn is homeomorphic to the order complex of the proper part of the partition lattice Πn.