  We introduce an algorithm for generating a random sequence of fragmentation
trees, which we call the ancestral branching algorithm. This algorithm builds
on the recursive partitioning structure of a tree and gives rise to an
associated family of Markovian transition kernels whose finite-dimensional
transition probabilities can be written in closed-form as the product over
partition-valued Markov kernels. The associated tree-valued Markov process is
infinitely exchangeable provided its associated partition-valued kernel is
infinitely exchangeable. We also identify a transition procedure on partitions,
called the cut-and-paste algorithm, which corresponds to a previously studied
partition-valued Markov process on partitions with a bounded number of blocks.
Specifically, we discuss the corresponding family of tree-valued Markov kernels
generated by the combination of both the ancestral branching and cut-and-paste
transition probabilities and show results for the equilibrium measure of this
process, as well as its associated mass fragmentation-valued and weighted
tree-valued processes.
