Permutation-valued Markov processes provide a convenient way to describe the genealogical structure of certain population models that allow immigration or mutation. Distinct cycles of the permutation correspond to binary branching trees that describe relationships among members of a particular family (or copies of an allele in the genetics setting), and the ordering of the cycles corresponds to families (or alleles) in the order of their appearance in the population. Building on the simple combinatorial structure of the Yule process with immigration, we describe the tree-valued processes that arise from linear birth and death processes and a population genetics model of Moran. This approach simplifies and explains much of the combinatorial structure of such processes, and relates genealogical (or time-reversed) processes with those running forward in time.
