The continuum random tree is the scaling limit of unlabeled unrooted trees

摘要

We show that the uniform unlabeled unrooted tree with n vertices and vertex degrees in a fixed set converges in the Gromov-Hausdorff sense after a suitable rescaling to the Brownian continuum random tree. This confirms a conjecture by Aldous (1991). We also establish Benjamini-Schramm convergence of this model of random trees and provide a general approximation result, that allows for a transfer of a wide range of asymptotic properties of extremal and additive graph parameters from Polya trees to unrooted trees.