Let G be a Galton-Watson tree, and for 0 u 1 let Gu be the subtree of G obtained by retaining each edge with probability u. We study the tree-valued Markov process (Gu; 0 u 1) and an analogous process (G u; 0 u 1) in which G 1 is a critical or subcritical Galton-Watson tree conditioned to be in nite. Results simplify and are further developed in the special case of Poisson( ) o spring distribut...

In this paper, we study the classification problem of discrete time and continuous time Markov processes with a tree structure. We first show some useful properties associated with the fixed points of a nondecreasing mapping. Mainly we find the conditions for a fixed point to be the minimal fixed point by using fixed point theory and degree theory. We then use these results to identify conditio...

We introduce two models for random trees with multiple states motivated by studies of trait dependence in the evolution of species. Our discrete time model, the multiple state ERM tree, is a generalization of Markov propagation models on a random tree generated by a binary search or 'equal rates Markov' mechanism. Our continuous time model, the multiple state Yule tree, is a generalization of t...