Covering a Tree by a Forest

Consider a tree T and a forest F . We discuss the following problems. Problem FVC: cover the vertices of T by a minimum number of copies of trees of F , such that every vertex of T is covered exactly once. Problem FEC: cover the edges of T with a minimum number of copies of trees of F , such that every edge of T is covered exactly once. F contains a one vertex (one edge) tree to always have a solution. We consider two versions of Problem FVC, ordered covers – Problem OFVC and unordered covers – Problem UFVC, and three versions of Problem FEC: consecutive covers – Problem CFEC, ordered covers – Problem OFEC and unordered covers – Problem UFEC. We describe polynomial time algorithms for Problems OFVC, UFVC and CFEC, and prove that Problems OFEC and UFEC are NP-complete.