What is the largest possible number of internal nodes in a red-black tree with black-height \(k\)? What is the smallest possible number?

Largest is a red-black tree with alternating levels of red nodes and black nodes which is a complete tree of height \(2k\) and so contains \(2^{2k} - 1\) internal nodes.

Smallest is a red-black tree with all black nodes which is a complete tree of height \(k\) and so contains \(2^{k} - 1\) internal nodes.