previous discussion about graph associahedra , generalized permutohedra and so on).

A map from Cayley functions to polytopes of Feynman diagrams {#sec2}
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An important property of Cayley functions is that we can directly read off the pole structures and consequently the sum of Feynman diagrams of their CHY formulas. We will see that the result provides an interesting map from any labelled tree to a polytope whose vertices correspond to Feynman diagrams. Note that these polytopes are only combinatoric, while in [@Arkani-Hamedsonghe] one can actually construct polytopes, [*e.g.*]{} the associahedron in Mandelstam space, whose canonical form (also see [@Arkani-Hamed:2017tmz] for definition) turns out to be the pushforward of the canonical form of ${\cal M}_{0,n}$ (also an associahedron).

CHY formulas for Cayley functions squared {#identicalCcc}
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Here we explain Theorem \[identicalCc\] in two steps. First we show what poles are allowed on the RHS of Theorem \[identicalCc\], and then provide a way to obtain all Feynman diagrams recursively . Then we give a classification of Cayley functions and also present detailed examples, for the two extreme cases, $C_n^H$ and $C_n^S$, as well as some other cases.

### The pole structure {#the-pole-structure .unnumbered}

The first result we present is the construction 