of its boundaries, for $r=0,1,\cdots,n-4$. In the end, each facet (dimension-$(n{-}4)$ face) corresponds to a pole.

So far, this map is realized by CHY formula of Cayley functions squared. However, we can abstractly view it as constructing a polytope from subgraph structure of a labelled tree: every dimension-$r$ face of the polytope corresponds a collection of $n-3-r$ compatible connected subgraphs of the tree.

For PT, the vertices of the corresponding polytope are all planar Feynman diagrams. The dual graph of each planar tree diagram is the triangulation of a $n$-gon, see figure \[triangulation\]. Two vertices are connected by an edge iff their triangulations differ by a single flip. It is well known that a polytope with such vertices is the so-called [*associahedron*](https://en.wikipedia.org/wiki/Associahedron) [@associahedron] living in $n-3$ dimensions, which we denote as ${\cal K}_{n-3}$ (in usual literature, it is called $K_{n-1}$). Therefore, we have mapped PT to an associahedron, and let’s give some explicit examples.

(x1) [ ]{}; at ($(x1)+(108:2)$) (x2) [ ]{}; at ($(x2)+(36:2)$) (x3) [ ]{}; at ($(x3)+(-36:2)$) (x4) [ ]{}; at ($(x4)+(-108:2)$) (x5) [ ]{};

(x1)node\[below left=0pt\]

(1); (a1) ; (a2) ; (a3) ; (5) ; (2) ; (3) ; (4) ; (1) node\[below=-2.5pt\][[4]{}]{} –(5)node\[below=-2.5pt\][[3]{}]{}; (a1) –(2)node\[above=-2.5pt\][[5]{}]{}; (a2) 