  Bergman complexes are polyhedral complexes associated to matroids. Faces of
these complexes are certain matroids, called matroid types, too. In order to
understand the structure of these faces we decompose matroid types into direct
summands. Ardila/Klivans proved that the Bergman Complex of a matroid can be
subdivided into the order complex of the proper part of its lattice of flats.
Beyond that Feichtner/Sturmfels showed that the Bergman complex can even be
subdivided to the even coarser nested set complex. We will give a much shorter
and more general proof of this fact. Generalizing formulas proposed by
Ardila/Klivans and Feichtner/Sturmfels for special cases, we present a
decomposition into direct sums working for faces of any of these complexes.
Additionally we show that it is the finest possible decomposition for faces of
the Bergman complex.
