By a theorem of Janson, the Wiener index of a random tree from a simply generated family of trees converges in distribution to a limit law that can be described in terms of the Brownian excursion. The family of unlabelled trees (rooted or unrooted), which is perhaps the most natural one from a graph-theoretical point of view, since isomorphisms are taken into account, is not covered directly by this theorem though. The aim of this paper is to show how one can prove the same limit law for unlabelled trees by means of generating functions and the method of moments. © 2011 Elsevier B.V. All rights reserved.
Wagner, S. (2012). On the Wiener index of random trees. Discrete Mathematics, 312(9), 1502–1511. https://doi.org/10.1016/j.disc.2011.05.008