On Graphs, Probability Theory and Cakes

Chef is viewing a simulation of creating a tree with $N$ vertices (numbered $1$ through $N$) which calculates mysterious coefficients for all the edges. Initially, there is a graph without edges, but there is a set $S$ containing $N-1$ edges (numbered $1$ through $N-1$) which should get added to the graph one by one. In this process, $N-1$ times, the following steps are performed: - An edge $e