Number of rooted labelled trees 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?
We don't consider the order of elements since what we are counting is the number of graphs on the vertex set $\{1,\ldots,n\}$ which are trees, that is, connected and having no cycles.

This is the usual graph-theoretic notion of tree. In computer science we often think of trees as rooted objects with ordered children, which is a similar but different concept.