The number of maximum matchings in a tree

TL;DR

This paper establishes bounds on the maximum number of maximum matchings in trees of a given size, characterizes the extremal trees, and improves existing bounds on maximal matchings.

Contribution

It provides a complete characterization of trees with the most maximum matchings and derives an exponential upper bound, refining previous results.

Findings

Maximum number of maximum matchings in a tree is at most O(1.391664^n)

Trees with the largest number of maximum matchings have a subtle structure

Improves bounds on the number of maximal matchings in trees

Abstract

We determine upper and lower bounds for the number of maximum matchings (i.e., matchings of maximum cardinality) $m(T)$ of a tree $T$ of given order. While the trees that attain the lower bound are easily characterised, the trees with largest number of maximum matchings show a very subtle structure. We give a complete characterisation of these trees and derive that the number of maximum matchings in a tree of order $n$ is at most $O(1.391664^n)$ (the precise constant being an algebraic number of degree 14). As a corollary, we improve on a recent result by G\'orska and Skupie\'n on the number of maximal matchings (maximal with respect to set inclusion).