We establish finite-sample guarantees for efficient proper learning of bounded-degree {\em polytrees}, a rich class of high-dimensional probability distributions and a subclass of Bayesian networks, a widely-studied type of graphical model. Recently, Bhattacharyya et al.(2021) obtained finite-sample guarantees for recovering tree-structured Bayesian networks, ie, 1-polytrees. We extend their results by providing an efficient algorithm which learns -polytrees in polynomial time and sample complexity for any bounded when the underlying undirected graph (skeleton) is known. We complement our algorithm with an information-theoretic sample complexity lower bound, showing that the dependence on the dimension and target accuracy parameters are nearly tight.