Tree-valued dynamics arise in applications in many areas such as computer science, machine learning, and phylogenetics, often in the context of Markov chain Monte Carlo inference. The immense size of phylogenetic trees has motivated a growing literature on asymptotic properties of such Markov chains and their scaling limits and continuum analogs.  In the 1990's David Aldous conjectured the existence of a scaling limit for a tree-valued Markov chain that can be thought of as the simple random walk on binary trees.  Despite significant interest, constructing the limiting process using traditional methods such as generators or martingale problems is challenging and, indeed, remains open.  In this talk we will discuss the recent resolution of Aldous's conjecture using a novel pathwise construction.  Along the way, we will discuss how some important intermediate processes we construct are related to integrable probability, specifically to up-down chains on branching graphs.