In this paper, we propose a new conjecture that the complete graph K4m+1 can be decomposed into copies of two arbitrary trees, each of size m, m≥ 1. To support this conjecture we prove that the complete graph K4cm+1 can be decomposed into copies of an arbitrary tree with m edges and copies of the graph H, where H is either a path with m edges or a star with m edges and where c is any positive integer. Further, we discuss related open problems. © 2021, The Author(s), under exclusive licence to Springer Japan KK, part of Springer Nature.