  In this note, we prove that the realization of associahedron discovered by
Arkani-Hamed, Bai, He, and Yun (ABHY) is a positive geometry for tree-level
S-matrix of scalars which have no color and which interact via cubic coupling.
More in detail, we consider diffeomorphic images of the ABHY associahedron. The
diffeomorphisms are linear maps parametrized by the right cosets of the
Dihedral group on n elements. The set of all the boundaries associated with
these copies of ABHY associahedron exhaust all the simple poles. We prove that
the sum over the diffeomorphic copies of ABHY associahedron is a positive
geometry and the total volume obtained by summing over all the dual
associahedra is proportional to the tree-level S matrix of (massive or
massless) scalar particles with cubic coupling. We then provide non-trivial
evidence that the projection of the planar scattering forms parametrized by the
Stokes polytope on these realizations of the associahedron leads to the
tree-level amplitudes of scalar particles, which interact via quartic coupling.
  Our results build on ideas laid out in our previous works, leading to further
evidence that a large class of positive geometries which are diffeomorphic to
the ABHY associahedron defines an ``amplituhedron" for a tree-level S matrix of
some local and unitary scalar theory. We also highlight a fundamental
obstruction in applying these ideas to discover positive geometry for the one
loop integrand when propagating states have no color.
