Limits of Random Trees. II

Local convergence of bounded degree graphs was introduced by Benjamini and Schramm. This result was extended further by Lyons to bounded average degree graphs. In this paper we study the convergence of random tree sequences with given degree distributions. Denote by (Formula presented.) the set of possible degree sequences of a labeled tree on n nodes. Let Dn be a random variable on (Formula presented.) and T(Dn) be a uniform random labeled tree with degree sequence Dn. We show that the sequence T(Dn) converges in probability if and only if (Formula presented.), where (Formula presented.) and D(1) is a random variable on (Formula presented.)