have the property that, similar to the case of PT factors , via CHY formula it maps to a sum of certain cubic tree graphs with coefficient $+1$ only : \[cubicsum\] C\_n({i,j})  [cubic tree graphs]{}: [ ]{}\_n C\_n\^2({i,j})=\_[g [“compatible with"]{} {i,j}]{} \_I 1 [s\_I]{}. As we will explain in sec 2, we say a Feynman diagram,$ g$, is “compatible with” $\{i,j\}$ if and only if the $n-3$ poles of $g$ correspond to $n-3$ mutually compatible connected subgraphs of the labelled tree $\{i,j\}$. We summarize this result as a theorem to be shown in sec \[sec2\]:

\[identicalCc\] \[identicalCc22\]

Our study of Cayley functions has been motivated by [@Arkani-Hamedsonghe], where the “pushforward" of differential forms on ${\cal M}_{0,n}$ to Mandelstam space ([*i.e.*]{} space spanned by some independent Mandelstam variables) has been considered. As explained in [@Arkani-Hamedsonghe], the pushforward of a half-integrand, which is a differential form in Mandelstam space , contains all the information of CHY formula of its square; in some sense, the discussions here are like the [*combinatoric*]{} version of the geometric story in [@Arkani-Hamedsonghe] (the idea of studying the combinatorics of “polytopes of Feynman diagrams" has been considered in [@JJCarrascoaafe] and also see [@GraphAssociahedra; @ConciniCProcesi; @AlexanderPostnikov; @AlexandePostnikovVictorReiner] for some 