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