Scaling limits of Markov branching trees and Galton-Watson trees conditioned on the number of vertices with out-degree in a given set

Abstract

We obtain scaling limits for Markov branching trees whose size is specified by the number of nodes whose out-degree lies in a given set. This extends recent results of Haas and Miermont in (Ann. Probab. 40 (2012) 2589-2666), which considered the case when the size of a tree is either its number of leaves or its number of vertices. We use our result to prove that the scaling limit of finite variance Galton-Watson trees conditioned on the number of nodes whose out-degree lies in a given set is the Brownian continuum random tree. The key to applying our result for Markov branching trees to conditioned Galton-Watson trees is a generalization of the classical Otter-Dwass formula. This is obtained by showing that the number of vertices in a Galton-Watson tree whose out-degree lies in a given set is distributed like the number of vertices in a Galton-Watson tree with a related offspring distribution.