On the maximal matchings of trees

An independent edge set of graph $G$ is a matching, and is maximal if it is not a proper subset of any other matching of $G$. The number of all the maximal matchings of $G$ is denoted by $\mathrm{\Psi }(G)$. In this paper, an algorithm to count $\mathrm{\Psi }(T)$ for a tree $T$ is given. We show that for any tree $T$ with $n$ vertices, $\mathrm{\Psi }(T)\ge \lceil \frac{n}{2}\rceil$, and the tree which obtained the lower bound is characterized.