Trees with the most subtrees -- an algorithmic approach

TL;DR

This paper presents an algorithmic method to identify trees with the maximum number of subtrees given specific order and degree sequence, providing a practical approach to extremal tree structures.

Contribution

It introduces an explicit algorithmic approach to find extremal trees with maximum subtrees based on order and degree sequence, extending previous theoretical results.

Findings

Algorithm explicitly constructs extremal trees

Provides insights into near-extremal trees

Connects subtree counts to number theory sequences

Abstract

When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in applications. The number of subtrees in the extremal cases constitute sequences which are of interest to number theorists. The structures which maximize or minimize the number of subtrees among general trees, binary trees and trees with a given maximum degree have been identified previously. Most recently, results of this nature are generalized to trees with a given degree sequence. In this note, we characterize the trees which maximize the number of subtrees among trees of a given order and degree sequence. Instead of using theoretical arguments, we take an algorithmic approach that explicitly describes the process of achieving an extremal tree from any random tree. The result also leads to some interesting questions and provides insight on finding the trees close to extremal and their numbers of subtrees.