Tree-valued Markov chains and Poisson-Galton-Watson distributions

The Poisson-Galton-Watson distribution on nite trees, and the related PGW 1 (1) distribution on innnite trees with one end, arise in several contexts, in particular as n ! 1 weak limits within various size-n combinato-rial models. We review this topic, introducingslick notation for describing such distributions. We then describe a family of continuous-time Markov chains whose marginal distributions are of Poisson-Galton-Watson type. Diierent chains in this family have connections with diierent parts of the extensive literature on tree and forest-valued chains (the random graph process, stochastic coalescence, spanning tree chains) and also illustrate the classical state-space-compactiication theory of continuous-time countable-state chains.