  Several hook summation formulae for binary trees have appeared recently in
the literature. In this paper we present an analogous formula for unordered
increasing trees of size r, which involves r parameters. The right-hand side
can be written nicely as a product of linear factors. We study two
specializations of this new formula, including Cayley's enumeration of trees
with respect to vertex degree. We give three proofs of the hook formula. One of
these proofs arises somewhat indirectly, from representation theory of the
symmetric groups, and in particular uses Kerov's character polynomials. The
other proofs are more direct, and of independent interest.
