Another Enumeration of Caterpillar Trees

TL;DR

This paper provides a new proof for counting non-isomorphic caterpillar trees with N vertices, using a natural vertex labelling and orbit-counting theorem, building on previous combinatorial enumeration methods.

Contribution

It introduces a novel proof technique employing integer-weighted vertex labelling and orbit-counting, offering a clearer combinatorial perspective on caterpillar enumeration.

Findings

Confirmed the enumeration formula for non-isomorphic caterpillars

Demonstrated the effectiveness of natural labelling and orbit-counting

Provided insights into symmetry and automorphisms of caterpillar trees

Abstract

A caterpillar tree is a connected, acyclic, graph in which all vertices are either a member of a central path, or joined to that central path by a single edge. In other words, caterpillar trees are the class of trees which become path graphs after removing all leaves. In 1973, F. Harary and A.J. Schwenk provided two proofs found in [1] which show that the number of non-isomorphic caterpillars with N vertices is given by the formula $2^{N-4} + \ 2^{ \lfloor \frac{N - 4}{2}\rfloor}$, where $\lfloor \ \rfloor$ denotes the floor function. The first proof follows from a special case of an application of P\'{o}lya's Enumeration theorem on graphs with integer-weighted vertices. The second proof proceeds through an appropriate edge labelling of the caterpillars. The proof presented here owes much of its insight to the first two, but has the benefit of utilizing a natural labelling for the caterpillars. We will proceed by labelling the vertices of the caterpillars with integer-weights, followed by an application of the orbit-counting theorem.