Can we maintain the black-heights of nodes in a red-black tree as attributes in the nodes of the tree without affecting the asymptotic performance of any of the red-black tree operations? Show how, or argue why not. How about maintaining the depth of nodes?
We can maintain the black heights, although we need to be careful with the various recoloring that's happening when we're going up the tree on insertion and deletion. Then it's quite simple to do it when rotations happen. I'm not going to show it, as I don't think it's that interesting.
Depth won't work, because if we remove the root, we may need to update the depth of all $n$ nodes.