Xor Decomposition

Chef has a tree with $N$ nodes (numbered $1$ through $N$) and a non-negative integer $x$. The nodes of the tree have non-negative integer weights; let's denote the weight of node $i$ by $a_i$. Next, let's define the *XOR value* of a tree as the bitwise XOR of weights of all its nodes. Chef wants to remove zero or more edges from his tree in such a way that each individual tree in the forest form