On q-functional equations and excursion moments

We analyse q-functional equations arising from tree-like combinatorial structures, which are counted by size, internal path length and certain generalisations thereof. The corresponding counting parameters are labelled by an integer k > 1. We show the existence of a joint limit distribution for these parameters in the limit of infinite size, if the size generating function has a square root as dominant singularity. The limit distribution coincides with that of integrals of (k− 1)th powers of the standard Brownian excursion. Our method yields a recursion for the moments of the joint distribution and admits an extension to other types of singularities.