  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
a positive integer k. 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-th powers of the standard Brownian excursion. Our
approach yields a recursion for the moments of the limit distribution. It can
be used to analyse asymptotic expansions of the moments, and it admits an
extension to other types of singularity.
