Consider a rooted labelled tree graph $\tau_n$ having a total of n vertices. The width function counts the number of vertices as a function of the distance to the root $\phi$. In this paper we compute large n asymptotic behavior of the width functions for two classes of tree graphs (both random and deterministic) of the following types: (i) Galton-Watson random trees $\tau_n$ conditioned on total progeny and (ii) a class of deterministic self-similar trees which include an "expected" Galton-Watson tree in a sense to be made precise. The main results include: (i) an extension of Aldous's theorem on "search-depth" approximations by Brownian excursion to the case of weighted Galton-Watson trees; (ii) a probabilistic derivation which generalizes previous results by Troutman and Karlinger on the asymptotic behavior of the expected width function and provides the fluctuation law; and (iii) width function asymptotics for a class of deterministic self-similar trees of interest in the study of river network data.
"Some width function asymptotics for weighted trees." Ann. Appl. Probab. 7 (4) 972 - 995, November 1997. https://doi.org/10.1214/aoap/1043862421