  The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for
every tree $T$ there exists a natural number $k(T)$ such that the following
holds: If $G$ is a $k(T)$-edge-connected simple graph with size divisible by
the size of $T$, then $G$ can be edge-decomposed into subgraphs isomorphic to
$T$. So far this conjecture has only been verified for paths, stars, and a
family of bistars. We prove a weaker version of the Tree Decomposition
Conjecture, where we require the subgraphs in the decomposition to be
isomorphic to graphs that can be obtained from $T$ by vertex-identifications.
We call such a subgraph a homomorphic copy of $T$. This implies the Tree
Decomposition Conjecture under the additional constraint that the girth of $G$
is greater than the diameter of $T$. As an application, we verify the Tree
Decomposition Conjecture for all trees of diameter at most 4.
