Random free trees and forests with constraints on multiplicities of vertices

We consider free (not rooted) trees with n labelled vertices whose multiplicities take values in some fixed subset A of non-negative integers such that A contains zero, A ≠ {0}, A ≠ {0, 1}, and the greatest common divisor of the numbers {k | k ∈ A} is equal to one. We find the asymptotic behaviour of the number of all these trees as n → ∞. Under the assumption that the uniform distribution is defined on the set of these trees, for the random variable , r ∈ A, which is equal to the number of vertices of multiplicity r in a randomly chosen tree, we find the asymptotic behaviour of the mathematical expectation and variance as n → ∞ and prove local normal and Poisson theorems for these random variables. For the case A = {0,1}, we obtain estimates of the number of all forests with n labelled vertices consisting of N free trees as n → ∞ under various constraints imposed on the function N = N(n). We find the asymptotic behaviour of the number of all forests of free trees with n vertices of multiplicities at most one. We prove local normal and Poisson theorems for the number of trees of given size and for the total number of trees in a random forest of this kind. We obtain limit distribution of the random variable equal to the size of the tree containing the vertex with given label.