We'll simply show that we don't have any choices to make, so that if there is a perfect matching, it's determined by the tree. Consider some leaf u. It must be matched with its parent, v. Vertex v might be adjacent to some other leaves, in which case there is no perfect matching. If not then remove the edge uv (since it must be in a perfect matching) and continue. We know that since the graph is acyclic, we'll always have a leaf. We continue the procedure until we end up with a perfect matching, or along the way find there isn't one. Since we never made any choices (every leaf must matched to its parent), there is at most one perfect matching.
perfect matching is matching with a covering . take set of those independent edges which would cover all the vertices here.