The Minimum Game

Alice and Bob have invented a new game. They have a rooted tree having $N$ nodes (rooted at node $1$), and the $i^{th}$ node has the value $A_i$ associated with it ($1$-indexed). The value associated with the root node is $0$. They start with a marker at the root node, and a score of $0$. Alice and Bob moves alternatively, with Alice moving first. In one move, they can move the pointer from a nod