Pruning Galton-Watson Trees and Tree-valued Markov Processes

Abstract: 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 ${{\cal G}(u)}$ by pruning Galton-Watson trees and an analogous process ${{\cal G}^*(u)}$ 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 ${{\cal G}(u)}$ run until its ascension time has a representation in terms of ${{\cal G}^*(u)}$. A similar result was obtained by Aldous and Pitman (1998) 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.