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 B ˜ ( K n ) of the graphical matroid of the complete graph K n is homeomorphic to the space of phylogenetic trees T n × R . This leads to a proof that the link of the origin in T n is homeomorphic to the order complex of the proper part of the partition lattice Π n .