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We introduce an algorithm for generating a random sequence of fragmentationtrees, which we call the ancestral branching algorithm. This algorithm buildson the recursive partitioning structure of a tree and gives rise to anassociated family of Markovian transition kernels whose finite-dimensionaltransition probabilities can be written in closed-form as the product overpartition-valued Markov kernels. The associated tree-valued Markov process isinfinitely exchangeable provided its associated partition-valued kernel isinfinitely exchangeable. We also identify a transition procedure on partitions,called the cut-and-paste algorithm, which corresponds to a previously studiedpartition-valued Markov process on partitions with a bounded number of blocks.Specifically, we discuss the corresponding family of tree-valued Markov kernelsgenerated by the combination of both the ancestral branching and cut-and-pastetransition probabilities and show results for the equilibrium measure of thisprocess, as well as its associated mass fragmentation-valued and weightedtree-valued processes.

Ancestral branching, cut-and-paste algorithms and associated tree and partition-valued processes