On the number of decomposable trees

A tree is called k-decomposable if it has a spanning forest whose components are all of size k. Analogously, a tree is called T -decomposable for a fixed tree T if it has a spanning forest whose components are all isomorphic to T . In this paper, we use a generating functions approach to derive exact and asymptotic results on the number of k-decomposable and T -decomposable trees from a so-called simply generated family of trees – we find that there is a surprisingly simple functional equation for the counting series of k-decomposable trees. In particular, we will study the limit case when k goes to∞. It turns out that the ratio of k-decomposable trees increases when k becomes large.