Scaling limits for a family of unrooted trees

We introduce weights on the unrooted unlabelled plane trees as follows: for each p = 1, let µp be a probability measure on the set of nonnegative integers whose mean is bounded by 1; then the µp-weight of a plane tree t is defined as ? µ_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 µ_p-weights. We prove that, under the assumption that the sequence of laws µ_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 Lévy trees, which have been introduced in Duquesne and Wang (2016+).