  We relate some features of Bruhat-Tits buildings and their compactifications
to tropical geometry. If G is a semisimple group over a suitable
non-Archimedean field, the stabilizers of points in the Bruhat-Tits building of
G and in some of its compactifications are described by tropical linear
algebra. The compactifications we consider arise from algebraic representations
of G. We show that the fan which is used to compactify an apartment in this
theory is given by the weight polytope of the representation and that it is
related to the tropicalization of the hypersurface given by the character of
the representation.
