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(1.391664n) (the precise constant being an algebraic number of degree 14). As a corollary, we improve on a recent result by Górska and Skupie on the number of maximal matchings (maximal with respect to set inclusion). © 2011 Elsevier B.V. All rights reserved.
Heuberger, C., & Wagner, S. (2011). The number of maximum matchings in a tree. Discrete Mathematics, 311(21), 2512–2542. https://doi.org/10.1016/j.disc.2011.07.028