  Planar arrays of tree diagrams were introduced as a generalization of Feynman
diagrams that enables the computation biadjoint amplitudes $m^{(k)}_n$ for
$k>2$ . In this follow-up work we investigate the poles of $m^{(k)}_n$ from the
perspective of such arrays. For general $k$ we characterize the underlying
polytope as a Flag Complex and propose a computation of the amplitude based
solely on the knowledge of poles, which number is drastically less than the
number of full arrays. As an example we first provide all the poles for the
cases $(k,n)=(3,7),(3,8),(4,8)$ and $(4,9)$ in terms of their generalized
Feynman diagrams. We then implement a simple compatibility criteria together
with an addition operation between arrays, and recover the full
collections/arrays recently presented for such cases. Along the way we
implement hard and soft kinematical limits, which provide a map between poles
in kinematic space and their combinatoric arrays. We use the operation to give
a proof of a previously conjectured combinatorial duality for arrays in $(k,n)$
and $(n-k,n)$. We also outline the relation to boundary maps of the
hypersimplex $\Delta_{k,n}$ and rays in the tropical Grassmannian
$\textrm{Tr}(k,n)$.
