Subtrees and BC-subtrees of maximum degree no more than k in trees

TL;DR

This paper develops recursive algorithms using generating functions to enumerate subtrees and BC-subtrees with maximum degree constraints in trees, providing detailed examples and discussing their densities.

Contribution

It introduces a novel recursive algorithmic approach for counting constrained subtrees and BC-subtrees in trees, expanding understanding of their enumeration and densities.

Findings

Algorithms successfully enumerate subtrees and BC-subtrees with degree constraints.

Detailed examples illustrate the recursive methods.

Discussion on densities of such subtrees among all subtrees.

Abstract

The subtrees and BC-subtrees (subtrees where any two leaves are at even distance apart) have been extensively studied in recent years. Such structures, under special constraints on degrees, have applications in many fields. Through an approach based on generating functions, we present recursive algorithms for enumerating various subtrees and BC-subtrees of maximum degree $\leq k$ in trees. The algorithms are illustrated through detailed examples. We also briefly discuss, in trees, the densities of subtrees (resp.~BC-subtrees) of maximum degree $\leq k(\geq 2)$ among all subtrees (resp.~BC-subtrees).