As we were for AVL trees, we're interested in what the red-black balancing rule guarantees about performance. Again, we'll simply state the results:
A red-black tree with n nodes has height at least log2 (n + 1) but no more than 2 * log2 (n + 1). A red-black tree with height h has at least pow (2, h / 2) - 1 nodes but no more than pow (2, h) - 1.
For comparison, an optimally balanced BST with n nodes has height ceil (log2 (n + 1)). An optimally balanced BST with height h has between pow (2, h - 1) and pow (2, h) - 1 nodes.
[Cormen 1990], lemma 14.1;
[Sedgewick 1998], property 13.8.