According to Cayley's formula, we have number of spanning trees on a complete graph as n^n-2 and number of labelled trees with n vertices as n^n-2
If the tree is rooted then in each tree we can choose a root in n ways so total number of rooted labelled trees is n^n-1.
Here when computing the total number for rooted labelled tree, why don't we consider the ordering of elements ?
For example if we have a regular graph with 3 vertices then one of the spanning tree is 1->2->3. Here each element can be root and we can have remaining elements at right or left. So why don't we consider the ordering (right or left) in the formula?