In many tree-structured parallel computations, the size and shape of a tree that represents a parallel computation is unpredictable at compile-time. The tree evolves gradually during the course of the computation. When such an application is executed on a static network, the dynamic tree evolution problem is to distribute the tree nodes to the processors of the network such that all the processors receive roughly the same amount of load and that communicating nodes are assigned to neighboring processors. The main contribution of the paper is to describe a simple randomwalk-based asymptotically optimal dynamic tree evolution algorithm on hypercubes and analyze the speed at which the performance ratio converges to the optimal. Our strategy is to prove that the Markov chain of the random walk on a hypercube is rapidly mixing.