We introduce extensions to an infinite-allele branching process that allows for multiple types to exist alongside labels. We consider a Markov branching process and general branching process under different assumptions, and show asymptotic results about the growth of the labels as well as the frequency spectrum. These results are motivated by two separate models. The Markov binary splitting results are motivated by a model of clonal evolution in cancer that considers the effect of both driver and passenger mutations on tumor growth. The general process has applications in viral reproduction and dynamics.

We extend the infinite-allele simple branching process of Griffiths and Pakes (1988) allowing the offspring to change types and labels. The model is developed and limit theorems are given for the growth of the number of labels of a specific type. We also discuss the asymptotics of the frequency spectrum. Finally, we present an application of the model's use in tumorigenesis.

Down through the years the Constitution has not always included all Americans — especially Blacks and women. No such charges have been leveled against the Religious Liberty clauses, though there have been occasions when their protections have been overridden or imperfectly applied. To remind us — as the Framers knew well — that “nothing human can be perfect,” we hear remarks from a distinguished Baptist pastor, an accomplished lawyer, and an eminent civil rights leader, Dr. Benjamin Hooks.