XOR Tree

There is a tree with $N$ vertices, rooted at vertex $1$. Vertex $i$ has the value $A_i$ written on it. Alice walks along this tree, starting at the root. When she is at vertex $u$: - If $u$ has no children, she stops. - Otherwise, suppose $u$ has $c$ children. She picks one of them at random (each one has a $\frac{1}{c}$ probability of being picked), and then moves to it. Alice also has a scor