Random trees with superexponential branching weights

We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors w(n) associated with the vertices of the tree and depending only on their individual degrees n. We focus on the case when w(n) grows faster than exponentially with n. In this case, the measures on trees of finite size N converge weakly as N tends to infinity to a measure which is concentrated on a single tree with one vertex of infinite degree. For explicit weight factors of the form w(n) = (( n - 1)!)(alpha) with alpha > 0, we obtain more refined results about the approach to the infinite volume limit.