Principal $\Gamma$-cone for a tree

Each orientation on a Dynkin graph $\Gamma$ defines a cone (in a certain real configuration space) which is further divided into chambers. We enumerate the number of chambers for two particular cones, which are called the pricipal $\Gamma$-cones and are attached to bipartite decompositions of $\Gamma$, by a use of hook length formulae. We prove that these pricipal cones are characterized by the maximality of the number of chambers in them.