On the number of $r$-matchings in a Tree
  An $r$-matching in a graph $G$ is a collection of edges in $G$ such that the distance between any two edges is at least $r$. A $2$-matching is also called an induced matching. In this paper, we estimate the maximum number of $r$-matchings in a tree of fixed order. We also prove that the $n$-vertex path has the maximum number of induced matchings among all $n$-vertex trees. 
Dong Yeap Kang, Jaehoon Kim, Younjin Kim, and Hiu-Fai Law
math.CO