  We consider Gibbs distributions on finite random plane trees with bounded
branching. We show that as the order of the tree grows to infinity, the
distribution of any finite neighborhood of the root of the tree converges to a
limit. We compute the limiting distribution explicitly and study its
properties. We introduce an infinite random tree consistent with these limiting
distributions and show that it satisfies a certain form of the Markov property.
We also study the growth of this tree and prove several limit theorems
including a diffusion approximation.
