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 $\{{\cal G}(u)\}$ 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.