  Based on the ideas in [CKP], we introduce the weighted analogue of the
branching rule for the classical hook length formula, and give two proofs of
this result. The first proof is completely bijective, and in a special case
gives a new short combinatorial proof of the hook length formula. Our second
proof is probabilistic, generalizing the (usual) hook walk proof of
Green-Nijenhuis-Wilf, as well as the q-walk of Kerov. Further applications are
also presented.
