Let ⊗ be an associative binary operator, and let a be an attribute maintained in each node of a red-black tree. Suppose that we want to include in each node x an additional attribute f such that x.f = x₁.a ⊗ x₂.a ⊗ …⊗ x_m.a, where x₁. x₂, . . . ,x_m is the inorder listing of nodes in the subtree rooted at x. Show how to update the f attributes in O(1) time after a rotation. Modify your argument slightly to apply it to the size attributes in order-statistic trees.