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 $\{\mathcal{G}(u)\}$ by pruning Galton-Watson trees and an analogous process $\{{\mathcal{G}}^{\ast }(u)\}$ 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 $\{\mathcal{G}(u)\}$ run until its ascension time has a representation in terms of $\{{\mathcal{G}}^{\ast }(u)\}$. 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.

References in corpus (1)

• Self-similar fragmentations derived from the stable tree II: splitting at nodes

Cited by in corpus (3)

• Local limits of conditioned Galton-Watson trees I: the infinite spine case
• Convergence of bi-measure R-tree and the pruning process
• Path-valued branching processes and nonlocal branching superprocesses