  Let $G$ be a complete graph with $n+1$ vertices. In a recent paper of the
authors, it is shown that the path trees of the graph play a special role in
the structure of the truncated powers and partition functions that are
associated with the graph. Motivated by the above, we take here a closer look
at the geometry of the simplicial cones associated with the graph, and the role
played by those simplicial cones that are generated by path trees. It is shown
that the latter form a basis for the linear space spanned by the former, and
that the representation of a general simplicial cone by path tree cones can be
deduced by examining partial orders induced by rooted trees. While the problem
itself is geometrical and its solution is combinatorial, the proofs rest with
multivariate spline theory.
