E - Counting is Fun

You are given a tree with $n$ nodes and a positive integer $c$. The $i$-th edge connects the nodes $u_i$ and $v_i$. Before the traversal begins, you must choose a permutation $p$ of $[1, 2, \ldots, n - 1]$ and label the $i$-th edge with $p_i$. These labels remain fixed throughout the traversal. You start from node $1$ with an integer $x = 1$. During the traversal, you can repeatedly perform o