On The Limiting Distributions of the Total Height On Families of Trees

A symbolic-computational algorithm, fully implemented in Maple, is described, that computes explicit expressions for generating functions that enable the efficient computations of the expectation, variance, and higher moments, of the random variable `sum of distances to the root', defined on any given family of rooted ordered trees (defined by degree restrictions). Taking limits, we confirm, via elementary methods, the fact, due to David Aldous, and expanded by Svante Janson and others, that the limiting (scaled) distributions are all the same, and coincide with the limiting distribution of the same random variable, when it is defined on labeled rooted trees.