http://www-math.mit.edu/~djk/18.310/18.310F04/counting_trees.html

Is this formula is approximation??

This problem solution "number of spanning trees in labeled complete graph is $$$n^{n-2}$$$", is called Cayley's Formula, which is famous in graph theory field.

There is a famous proof by Heinz Prüfer (1918) which uses Prüfer Code. But we can able to count number of spanning trees in any undirected simple graph $$$G$$$ via determinant of matrix. This way is called Kirchhoff's Theorem.