Scoring Tree

Alice and Bob are playing a game on a tree with $N$ nodes and $N - 1$ edges. They take turns alternately, with **Alice going first**. On each turn, the current player selects any node that has not been chosen yet. The game ends when all $N$ nodes have been selected. After the game ends, Alice's **reward** is calculated as follows: A node $w$ selected by Bob is called *captured* if there exist t