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.