Scaling limits for a family of unrooted trees

We introduce weights on the unrooted unlabelled plane trees as follows: for each p >= 1, let mu(p) be a probability measure on the set of nonnegative integers whose mean is bounded by 1; then the mu(p)-weight of a plane tree t is defined as Pi mu(p)(degree(v) - 1), where the product is over the set of vertices v of t. We study the random plane tree T-p which has a fixed diameter p and is sampled according to probabilities proportional to these mu(p)-weights. We prove that, under the assumption that the sequence of laws mu(p), p >= 1, belongs to the domain of attraction of an infinitely divisible law, the scaling limits of (T-p, p >= 1) are random compact real trees called the unrooted Levy trees, which have been introduced in (2016+).