Red-Black Tree

An extended rooted binary tree satisfying the following conditions:

1. Every node has two children, each colored either red or black.

2. Every tree leaf node is colored black.

3. Every red node has both of its children colored black.

4. Every path from the root to a tree leaf contains the same number (the "black-height") of black nodes.

Let be the number of internal nodes of a red-black tree. Then the number of red-black trees for , 2, ... is 2, 2, 3, 8, 14, 20, 35, 64, 122, ... (OEIS A001131). The number of trees with black roots and red roots are given by OEIS A001137 and OEIS A001138, respectively.

Let be the generating function for the number of red-black trees of black-height indexed by the number of tree leaves. Then

(1)

where . If is the generating function for the number of red-black trees, then

(2)

(Ruskey). Let be the number of red-black trees with tree leaves, the number of red-rooted trees, and the number of black-rooted trees. All three of the quantities satisfy the recurrence relation

(3)

where is a binomial coefficient, , for , , for , and for (Ruskey).