Red-Black Tree

An extended rootedbinary 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, ... (Sloane's A001131). The number of trees with black roots and red roots are given by Sloane's A001137 and Sloane's 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).

1,每个节点有两个子节点,要么是红色要么是黑色
2,每个叶节点都为黑色
3,每个红节点都有两个黑色的子节点
4,从根到叶的每一条路径都包括相同的黑节点.如上图的左边两级,右边有三级
根节点的左节点的两个子节点都有黑色,那为什么它不为红色呢?因为它要满足第四点.所以为黑色