{2,3}, {3,4})=1[\_[12]{} \_[23]{} \_[34]{}]{}, C\_5({1,2}, {1,3}, {1,4})=1 [\_[12]{}\_[13]{} \_[14]{}]{}.

Since Cayley functions in this $\sigma_n \to \infty$ frame are in one-to-one correspondence with $(n{-}1)$-pt labelled trees, there are exactly $(n{-}1)^{n{-}3}$ of them. A basic question we are interested in is how many different classes of Cayley functions there are; two Cayley functions are said to be topologically equivalent if and only if their tree graphs can be brought to the same shape, which are just relabelling of each other, for example the there are two classes of Cayley functions for $n=5$. Generally for any $n$, there are always these two extreme classes, given by the so-called Hamiltonian graph (a line) and star graph (a star-shaped tree, and we can choose the center to be [*e.g.*]{} 1) respectively (see figure \[S graph\]) C\^[H]{}\_n=\_[a=1]{}\^[n-2]{} 1 [\_[(a), (a[+]{}1)]{}]{}, C\^[S]{}\_n=\_[a=2]{}\^[n-1]{} 1 [\_[1, a]{}]{}. The former is nothing but the Parke-Taylor factor ${\rm PT}(\alpha(1), \alpha(2), \cdots, \alpha(n{-}1), n)$ in the ${\rm SL}(2, {\mathbb C})$-fixed form, , while the latter is totally symmetric in labels $2,3,\ldots, n{-}1$.

For $n\geq 6$, we have new classes that are in between these two extremes, see examples in sec.\[sec2\]. The number of distinct classes is nothing but the number of 