On the number of nonisomorphic subtrees of a tree

TL;DR

This paper establishes an upper bound on the number of nonisomorphic subtrees in a tree of order n, demonstrating that the maximum is proportional to 5^{n/4} and is tight, with similar results for rooted subtrees.

Contribution

It provides the first tight asymptotic bound on the number of nonisomorphic subtrees of a tree, including rooted variants.

Findings

Maximum number of nonisomorphic subtrees is O(5^{n/4})

Bound is proven to be tight

Results extend to rooted subtrees

Abstract

We show that a tree of order $n$ has at most $O(5^{n/4})$ nonisomorphic subtrees, and that this bound is best possible. We also prove an analogous result for the number of nonisomorphic rooted subtrees of a rooted tree.