unlabelled trees, which equals $1,2,3,6,11,23,47,\cdots$ for $n{=}4,5,6,7,8,9,10,\cdots$ respectively (see [A000055](https://oeis.org/search?q=1%2C2%2C3%2C6%2C11%2C23%2C47%2C106
&sort=&language=english&go=Search)  [@A000055] ) . We will see that these classes play an important role in our following discussions.

Although we have defined Cayley functions in $\sigma_n\to \infty$ frame, it is straightforward to recover the ${\rm SL}(2,{\mathbb C})$ redundancy which results in the unique covariant form for each $C$: \_n({i,j}):=C\_n({i,j}) \_[1,n]{}\^[v\_1-2]{}\_[2,n]{}\^[v\_2-2]{}\_[n[-]{}1,n]{}\^[v\_[n[-]{}1]{}-2]{}, where $v_a$ denotes the valency of vertex $a=1,2,\cdots, n{-}1$ in the labelled tree, and by definition, $1\leq v_a \leq n{-}3$. It is easy to check that in ${\bf C}$ every $\sigma_a$ appears exactly twice in the denominator, including $\sigma_n$; this is because $\sum_{a=1}^{n-1} (v_a-2)=2(n-2)-2(n-1)=-2$, thus ${\bf C}_n$ given above is indeed ${\rm SL}(2,{\mathbb C})$ covariant with weight 2. The covariant form of $C^H_n={\rm PT}_n$ is of course ${\bf PT}_n$ in , and that for $C^S_n$ is given by \[cccova\] [**C**]{}\^S\_n=\_[a=2]{}\^[n[-]{}1]{} 1 [\_[1,a]{}]{} \_[1,n]{}\^[n[-]{}4]{}\_[a=2]{}\^[n[-]{}1]{} \_[a,n]{}\^[-1]{} =.

The Cayley functions generalize Parke-Taylor factors in an interesting way; it is well known that via partial-fraction identities, they can be redued to PT’s (see sec \[sec3p1\]), but we find it intriguing and useful to study these functions directly, in the context of CHY formula and string integrals. As we will see shortly, Cayley functions 