  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).
