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3 for any two distinct vertices x, y in S. The sparsity γ(
doi:10.1016/0012-365X(95)00041-T 
Copyright © 1996 Published by Elsevier Science B.V.
The enumeration of trees with and without given limbs
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Taojun Lu
Received 5 January 1993;
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.
