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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 to 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 $w_n=((n-1)!)^α$ with $α>0$ we obtain more refined results about the approach to the infinite volume limit.