  The biadjoint scalar partial amplitude, $m_n(\mathbb{I},\mathbb{I})$, can be
expressed as a single integral over the positive tropical Grassmannian thus
producing a Global Schwinger Parameterization. The first result in this work is
an extension to all partial amplitudes $m_n(\alpha,\beta)$ using a limiting
procedure on kinematic invariants that produces indicator functions in the
integrand. The same limiting procedure leads to an integral representation of
$\phi^4$ amplitudes where indicator functions turn into Dirac delta functions.
Their support decomposes into $\textrm{C}_{n/2-1}$ regions, with $\textrm{C}_q$
the $q^{\rm th}$-Catalan number. The contribution from each region is
identified with a $m_{n/2+1}(\alpha,\mathbb{I})$ amplitude. We provide a
combinatorial description of the regions in terms of non-crossing chord
diagrams and propose a general formula for $\phi^4$ amplitudes using the
Lagrange inversion construction. We start the exploration of $\phi^p$ theories,
finding that their regions are encoded in non-crossing $(p-2)$-chord diagrams.
The structure of the expansion of $\phi^p$ amplitudes in terms of $\phi^3$
amplitudes is the same as that of Green functions in terms of connected Green
functions in the planar limit of $\Phi^{p-1}$ matrix models. We also discuss
possible connections to recent constructions based on Stokes polytopes and
accordiohedra.
