Universal height and width bounds for random trees

Abstract

We prove non-asymptotic stretched exponential tail bounds on the height of a randomly sampled node in a random combinatorial tree, which we use to prove bounds on the heights and widths of random trees from a variety of models. Our results allow us to prove a conjecture and settle an open problem of Janson (https://doi.org/10.1214/11-PS188), and nearly prove another conjecture and settle another open problem from the same work (up to a polylogarithmic factor). The key tool for our work is an equivalence in law between the degrees along the path to a random node in a random tree with given degree statistics, and a random truncation of a size-biased ordering of the degrees of such a tree. We also exploit a Poissonization trick introduced by Camarri and Pitman (https://doi.org/10.1214/EJP.v5-58) in the context of inhomogeneous continuum random trees, which we adapt to the setting of random trees with fixed degrees. Finally, we propose and justify a change to the conventions of branching process nomenclature: the name "Galton-Watson trees" should be permanently retired by the community, and replaced with the name "Bienaym\'e trees".