SPDE Approximation for Random Trees

We consider the genealogy tree for a critical branching process conditioned on non-extinction. We enumerate vertices in each generation of the tree so that for each two generations one can define a monotone map describing the ancestor--descendant relation between their vertices. We show that under appropriate rescaling this family of monotone maps converges in distribution in a special topology to a limiting flow of discontinuous monotone maps which can be seen as a continuum tree. This flow is a solution of an SPDE with respect to a Brownian sheet.