Projectively consistent random walks on a space of trees with integer edge weights

Consider a randomly changing binary tree in which, at each step, you delete a random leaf and then grow a new leaf in a random location on the tree. This is applied in algorithms for phylogenetic inference. In 2000, Aldous conjectured that this process should have a continuum analogue, which would be a continuum random tree-valued diffusion. We will discuss a family of projectively consistent stochastic processes that are projections of this tree-valued process, as well as how these projections relate to Aldous's conjecture.