In this talk we first introduce another way of constructing new matroids from a given one M, by minimizing the linear functional w \cdot x over x in the matroid polytope of M, for a fixed w in R^n. We then use this to define the Bergman fan of a given matroid, which are spaces of weights w whose corresponding matroids do not contain loops. To understand the topology of the Bergman fan, we introduce a combinatorial gadget called phylogenetic trees which can be used to parametrize the Bergman fan, and give a nice subdivision of it.