Tree Game

There are $N$ cities numbered from $1$ to $N$, connected by $(N-1)$ roads such that every city is reachable from every other city (they form a tree structure). The $i^{th}$ city has a population $P_i$. Alice and Bob are initially standing at cities $A$ and $B$ respectively. They decide to play a game, such that, at every turn: - First, Alice makes a move by moving to some adjacent city **unvisite