2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/209294We 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.35 pages, 1 figure. New introduction, typographical errors correctedCombinatoricsProbability05A15, 39A13, 60C05,82B41text