  Recently, the accordiohedron in kinematic space was proposed as the positive
geometry for planar tree-level scattering amplitudes in the $\phi^p$ theory
\cite{Raman:2019utu}. The scattering amplitudes are given as a weighted sum
over canonical forms of some accordiohedra with appropriate weights. These
weights were determined by demanding that the weighted sum corresponds to the
scattering amplitudes. It means that we need additional data from the quantum
field theory to compute amplitudes from the geometry. It has been an important
problem whether scattering amplitudes are completely obtained from only the
geometry even in this $\phi^p$ theory. In this paper, we show that these
weights are completely determined by the factorization property of the
accordiohedron. It means that the geometry of the accordiohedron is enough to
determine these weights. In addition to this, we study one-parameter recursion
relations for the $\phi^p$ amplitudes. The one-parameter "BCFW"-like recursion
relation for the $\phi^3$ amplitudes was obtained from the triangulation of the
ABHY-associahedron \cite{Arkani-Hamed:2017tmz}. After this, a new recursion
relation was proposed from the projecting triangulation of the generalized
ABHY-associahedron in \cite{Arkani-Hamed:2019vag, Yang:2019esm}. We generalize
these one-parameter recursion relations to the $\phi^p$ amplitudes and
interpret as triangulations of the accordiohedra.
