Title: Limits of finite trees

Abstract: Motivated by the work of Lovász, Szegedy and others on the convergence and limits of dense graph sequences, we investigate the convergence and limits of finite trees with respect to sampling in normalized distance. In particular, we say that a sequence of finite trees converges if choosing r vertices uniformly at random, the distribution of the matrix describing their pairwise distances (scaled appropriately) weakly converge for all r. We introduce dendrons (a notion based on separable real trees) and show that the sampling limits of finite trees are exactly the dendrons. We also prove that the limit dendron is unique. The main proof technique is ultraproduct of metric and measure spaces.

This is joint work with Gábor Elek.