Generalized Leech Trees

In 1975, Leech introduced the problem of labeling the edges of a tree with distinct positive integers so that the sums along distinct paths in the tree were distinct, and the set of such path-sums were consecutive starting with one. We generalize this problem to labelings from arbitrary finite Abelian groups, with a particular focus on direct products of the additive group of \( \mathbb{Z}_2 \).