For $\displaystyle n \geq 2 $ , let u be a fixed vertex of the complete graph $\displaystyle K_{n} $. Compute the number $\displaystyle \tau_{u}(n) $ of spanning trees of $\displaystyle K_{n} $ that contain the vertex u as a leaf. Use this to show that the probability a vertex in a tree on n vertices is a leaf is approximately $\displaystyle 1/e $ where $\displaystyle e = 2.71828182... $ is the base number for the natural logarithm.