LetH=(VH,EH) be a graph, and letkbe a positive integer. A graphG=(VG,EG) isH-coverable with overlap kif there is a covering of the edges ofGby copies ofHsuch that no edge ofGis covered more thanktimes. Denote by overlap(H,G) the minimumkfor whichGisH-coverable with overlapk. Theredundancyof a covering that usestcopies ofHis (t|EH|-|EG|)/|EG|. Our main result is the following: IfHis a tree onhvertices andGis a graph with minimum degreeδ(G)≥(2h)10+C, whereCis an absolute constant, then overlap(H,G)≤2. Furthermore, one can find such a covering with overlap 2 and redundancy at most 1.5/δ(G)0.1. This result is tight in the sense that for every treeHonh≥4 vertices and for every functionf, the problem of deciding if a graph withδ(G)≥f(h) has overlap(H,G)=1 is NP-complete.