and at the lowest level you can interchange where they live and you can always make x and y appear together and put W over here and so there now have the same errand it's effectively the same code you just move things around at the bottom so x and y have the same current then every optimal prefix 3 will have x + y together at the lowest level with the same parent so that's what we're out now the steps we've made allow us to include this line over here no matter the tree structure the additional cost of having x and y rather than just