A linear algorithm for computing of a minimum weight maximal induced matching in an edge-weighted tree

An induced matching $M$ of a graph $G$ is a set of pairwise non-adjacent edges such that their end-vertices induce a 1-regular subgraph. An induced matching $M$ is maximal if no other induced matching contains $M$. The Minimum Induced Matching problem asks for a minimum maximal induced matching in a given graph. The problem is known to be NP-complete. Here we show that, if $G$ is a tree then this problem can be solved in linear time.