Red-Black-Trees - Red-Black Trees Contents Introduction...

Red-Black Trees
Contents
• Introduction
• Red-Black Tree Operations
• insert
• delete
• Summary
Introduction
Recall that, for binary search trees, although the average-case times for the lookup, insert, and delete
methods are all O(log N), where N is the number of nodes in the tree, the worst-case time is O(N). We
can guarantee O(log N) time for all three methods by using a balanced tree -- a tree that always has
height O(log N)-- instead of a binary search tree.
A number of different balanced trees have been defined, including AVL trees, 2-4 trees, and B trees. You
might learn about the first two in an algorithms class and the third in a database class. Here we will look
at yet another kind of balanced tree called a red-black tree.
The important idea behind all of these trees is that the insert and delete operations may restructure the
tree to keep it balanced. So lookup, insert, and delete will always be logarithmic in the number of nodes
but insert and delete may be more complicated than for binary search trees.
A red-black tree is a binary search tree in which
• each node has a color (red or black) associated with it (in addition to its key and left and right
children)
• the following 3 properties hold:
1. (root property) The root of the red-black tree is black
2. (red property) The children of a red node are black.
3. (black property) For each node with at least one null child, the number of black nodes on
the path from the root to the null child is the same.
An example of a red-black tree is shown below:
Operations on a Red-Black Tree
As with the binary search tree, we will want to be able to perform the following operations on red-black
trees:
• insert a key value (insert)