  In this paper, we show that for every tree $T$, there exists a positive
integer $l_T$ such that every $50|E(T)|^2$-edge-connected simple graph $G$ of
size divisible by $|E(T)|$ with minimum degree at least $l_T$ has an
edge-decomposition into isomorphic copies of $T$. In particular, we push this
quadratic bound down to a linear bound for trees with bounded number of
non-leaf vertices. This result is an improvement of a result due to Bensmail,
Harutyunyan, Le, Merker, and Thomass\'e (2017) who gave an exponential upper
bound on the needed edge-connectivity.
