The number of maximum matchings in a tree

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 the 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 ( 1.39166 4 n ) (the precise constant being an algebraic number of degree 14). As a corollary, we improve on a recent result by Górska and Skupień on the number of maximal matchings (maximal with respect to set inclusion).