  We construct polyhedral families of valuations on the branching algebra of a
morphism of reductive groups. This establishes a connection between the
combinatorial rules for studying a branching problem and the tropical geometry
of the branching algebra. In the special case when the branching problem comes
from the inclusion of a Levi subgroup or a diagonal subgroup, we use the dual
canonical basis of Lusztig and Kashiwara to build toric deformations of the
branching algebra.
