The distribution of nodes of given degree in random trees

Let Tn denote the set of unrooted unlabeled trees of size n and let k ≥ 1 be given. By assuming that every tree of Tn is equally likely it is shown that the limiting distribution of the number of nodes of degree k is normal with mean value ∼ μkn and variance ∼ σ 2 k n with positive constants μk and σk. Besides, the asymptotic behavior of μk and σk for k → ∞ as well as the corresponding multivariate distributions are derived. Furthermore, similar results can be proved for plane trees, for labeled trees, and for forests.