On isomorphism classes of leaf-induced subtrees in topological trees

TL;DR

This paper investigates the diversity of leaf-induced subtrees in topological trees, providing formulas and asymptotic estimates for their counts, and characterizing trees with minimal such diversity.

Contribution

It introduces new formulas and asymptotic estimates for counting nonisomorphic leaf-induced subtrees in various classes of topological trees, and characterizes trees with minimal counts.

Findings

Stars and binary caterpillars have the fewest nonisomorphic leaf-induced subtrees.

Derived closed and recursive formulas for d-ary caterpillars and complete d-ary trees.

Provided an asymptotic formula for the number of nonisomorphic leaf-induced subtrees in complete d-ary trees.

Abstract

A subtree can be induced in a natural way by a subset of leaves of a rooted tree. We study the number of nonisomorphic such subtrees induced by leaves (leaf-induced subtrees) of a rooted tree with no vertex of outdegree 1 (topological tree). We show that only stars and binary caterpillars have the minimum nonisomorphic leaf-induced subtrees among all topological trees with a given number of leaves. We obtain a closed formula and a recursive formula for the families of $d$-ary caterpillars and complete $d$-ary trees, respectively. An asymptotic formula is found for complete $d$-ary trees using polynomial recurrences. We also show that the complete binary tree of height $h>1$ contains precisely $\lfloor 2(1.24602...)^{2^h}\rfloor$ nonisomorphic leaf-induced subtrees.