Decomposing graphs into paths and trees

In [Bensmail, J., A. Harutyunyan, T.-N. Le and S. Thomassé, Edge-partitioning a graph into paths: beyond the Barát-Thomassen conjecture, arXiv preprint arXiv:1507.08208 (2015)], the authors conjecture that for a fixed tree T, the edge set of any graph G of size divisible by size of T with sufficiently high degree can be decomposed into disjoint copies of T, provided that G is sufficiently highly connected in terms of maximal degree of T. In [Bensmail, J., A. Harutyunyan, T.-N. Le and S. Thomassé, Edge-partitioning a graph into paths: beyond the Barát-Thomassen conjecture, arXiv preprint arXiv:1507.08208 (2015)], the conjecture was proven for trees of maximal degree 2 (i.e., paths). In particular, it was shown that in the case of paths, the conjecture holds for 24-edge-connected graphs. We improve this result showing that 3-edge-connectivity suffices, which is best possible. We disprove the conjecture for trees of maximum degree greater than two and prove a relaxed version of the conjecture that concerns decomposing the edge set of a graph into disjoint copies of two fixed trees of coprime sizes.