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 sub(n)associated with the vertices of the tree and depending only on their individual degrees n. We focus on the case when w sub(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 sub(n)= ((n - 1)!) alpha with alpha > 0, we obtain more refined results about the approach to the infinite volume limit.