Abstract :
A limb of a tree is the union of one or more branches at a vertex in the tree, where a branch of a tree at a vertex is a maximal subtree containing the given vertex as an end-vertex. In this note, we first consider the enumeration of trees (undirected, oriented or mixed) with forbidden limbs. The enumeration result for trees with a single forbidden limb proves directly that the number of trees (undirected, oriented or mixed) with a forbidden limb is independent of the structure of the limb (in the undirected case, Schwenk has given a bijective proof). We further extend the method to enumerate trees with a number of forbidden limbs. Finally we enumerate trees with respect to the number of vertices and the number of edge-disjoint limbs that are isomorphic to a given limb. Many examples are given for illustration.
