Scaling limits for a family of unrooted trees

We introduce weights on the unrooted unlabelled plane trees as follows: let $\mu$ be a probability measure on the set of nonnegative integers whose mean is no larger than $1$; then the $\mu$-weight of a plane tree $t$ is defined as $\Pi \, \mu (degree (v) -1)$, where the product is over the set of vertices $v$ of $t$. We study the random plane tree with a fixed diameter $p$ sampled according to probabilities proportional to these $\mu$-weights and we prove that, under the assumption that the sequence of laws $\mu_p$, $p\! \geq \! 1$, belongs to the domain of attraction of an infinitely divisible law, the scaling limits of such random plane trees are random compact real trees called the unrooted Levy trees, which have been introduced in Duquense & Wang.