Move the Coins 2

This is a sequel to the problem ["Move the Coins"](https://www.hackerrank.com/challenges/move-the-coins/problem) with a couple of twists! Alice and Bob are playing a board game. The board is a tree $T_{\textit{board}}$ with $N$ nodes (numbered $1$ through $N$). The game involves moving some coins around the board. For a node $R$ ($1 \le R \le N$), let's define a game $g(T_{\textit{board}}, R)$