While reading this paper, I stumbled over the statement (1, 8) about linear independence in trees.
Just to make sure, that I understand it correctly: He means conical independence and forces $a_{i, k} \geq 0$?.
Linear independence as soon as the number of vertices $n$ is larger than the dimension should be false.
No, this is linear independence.
A tree always has $n$ vertices and $n-1$ edges, so here we are talking about the linear independence of $n-1$ elements of an $n$-dimensional space. This should always be possible; there is no dimension argument against it.
Another way to phrase the argument for proving this linear independence is by induction on the size of the tree. Suppose we have an $n$-vertex tree and vertex $s$ is a leaf with neighbor $t$. Then if $$ \sum_{vw \in E(T), v<w} a_{vw}(x_v - x_w) = 0 $$ the coefficient of $x_s$ is $a_{st}$ alone, so $a_{st} = 0$. Therefore $$ \sum_{vw \in E(T - s), v<w} a_{vw} (x_v - x_w) = 0 $$ and we have reduced the problem to one about the $(n-1)$-vertex tree $T-s$.