In a fully balanced (symmetric) BST with n nodes, the root has rank (n/2), meaning the left and right subtrees have equal size. This properly will not always hold for a BST that is a valid red-black tree, even though we have proved that red-black trees are reasonably height-balanced.

Show how low (or if you prefer, symmetrically, how high) the rank of the root can be in a red-black tree. Use theta-notation for your answer, otherwise it will be a bit too technical at the final steps.