Pruning Galton-Watson trees and tree-valued Markov processes

We present a new pruning procedure on discrete trees by adding marks on the nodes of trees. This procedure allows us to construct and study a tree-valued Markov process {G(u)} by pruning Galton-Watson trees and an analogous process {G^*(u)} by pruning a critical or subcritical Galton-Watson tree conditioned to be infinite. Under a mild condition on offspring distributions, we show that the process {G(u)} run until its ascension time has a representation in terms of {G^*(u)}. A similar result was obtained by Aldous and Pitman (Ann. Inst. H. Poincaré Probab. Statist. 34 (1998) 637-686) in the special case of Poisson offspring distributions where they considered uniform pruning of Galton-Watson trees by adding marks on the edges of trees.