Let $T$ be a rooted tree with $m$ leaves. Label every edge with a label of the form $x_i$ or $-x_i$, for some letter $x_i$. For each leaf in the tree, consider the formal linear combination $v$ obtained by summing the labels in the edges from the root to the leaf. What can we say about the dimension of the space $V$ spanned by such $v$?

If all $x_i$'s are distinct, then clearly $\dim V = m$. Say that each $x_i$ appears at most $k$ times. Can one then say that $\dim V \geq m/k - r$, where $r$ is the maximal number of disjoint paths of length $>0$ in the tree such that no inner node of any path is an ancestor of any vertex in any other path?