Consider that the game can be modeled as a variant of the "take-away" game, where each move splits the tree into smaller subtrees. Think about how the bitwise AND operation affects the possible moves and outcomes.

Observe that the bitwise AND of a subtree's values can only decrease or stay the same as the subtree grows. This suggests that smaller subtrees (e.g., single nodes) might be strategically important. Explore how selecting single nodes first could influence the game's outcome.

Realize that the problem can be reduced to evaluating the game's outcome for every possible subtree rooted at each node. Use dynamic programming or memoization to compute the optimal moves for each subtree, considering the XOR of all possible AND values Alice and Bob can achieve. The key is to determine whether the root subtree's optimal outcome favors Alice or Bob.