compatible with ordering $\alpha$, $T(\alpha)$ (all planar cubic tree graphs with external legs in the ordering $\alpha$): \[map\] [**PT**]{}  [planar cubic tree graphs]{}:\_n [**PT**]{}()\^2=\_[g [compatible with]{} ]{}\^[[Cat]{}\_[n[-]{}2]{}]{} \_I 1 [s\_I]{}, where ${\rm Cat}_{n{-}2}$ denotes the [Catalan number](https://en.wikipedia.org/wiki/Catalan_number) $1, 2, 5, 14, 42, 132, \ldots$ for $n=3,4,5,6,7,8, \ldots$  [@catalan]. In this paper, we will study a new class of half integrands, which largely generalize Parke-Taylor factors and maps to collections of Feynman diagrams. In addition, they naturally appear in superstring disk integrals and we will discuss their applications in that direction as well.

Cayley functions and the map to cubic Feynman graphs
----------------------------------------------------

The main character of our story is a new class of half-integrands that we call [ Cayley functions]{}. Before proceeding, let us discuss a convenient way of fixing ${\rm SL}(2,{\mathbb C})$ in CHY formulas. Recall that we need to fix three punctures: we can always choose $\sigma_n\to \infty$, and fix any two more punctures at finite positions, [*e.g.*]{} $\sigma_1=0$, $\sigma_{n{-}1}=1$ which won’t be necessary to explicitly write down. With $\s_n\rightarrow \infty$, ${\rm SL}(2,{\mathbb C})$-fixed CHY formula reads \_n=\_n [I]{}\_n, \_n:=\_[a=2]{}\^[n[-]{}1]{} \_a(\_[ba]{} ), where the four infinite factors containing $\sigma_n$ in ${\mathrm{d}}{\bm \mu}_n$ cancel against those in ${\bf I}_n$ 