1. Red-Black Tree Neil Tang 02/07/2008
CS223 Advanced Data Structures and Algorithms

2. CS223 Advanced Data Structures and Algorithms
Class Overview
Definition
Tree height
Rotation and color flip
Insert
Delete
CS223 Advanced Data Structures and Algorithms

3. CS223 Advanced Data Structures and Algorithms
Definition
A red-black tree is a special binary search tree in which
every node is either red or black;
the root is black;
every leaf (null) is black;
if a node is red, then both its children are black;
every simple path from a node to a descendant leaf contains the same number of black nodes.
CS223 Advanced Data Structures and Algorithms

4. CS223 Advanced Data Structures and Algorithms
Height
Theorem: A red-black tree with N nodes has height at most 2log(N+1). (pp.274 Cormen’s book)
Experiments suggests that the average red-black tree is about as deep as an average AVL tree.
The rotation happens less frequently.
CS223 Advanced Data Structures and Algorithms

5. CS223 Advanced Data Structures and Algorithms
An Example
CS223 Advanced Data Structures and Algorithms

6. CS223 Advanced Data Structures and Algorithms
Rotation if S is Black
CS223 Advanced Data Structures and Algorithms

7. CS223 Advanced Data Structures and Algorithms
Color Flip if S is red
CS223 Advanced Data Structures and Algorithms

8. CS223 Advanced Data Structures and Algorithms
Example: Insert 45
CS223 Advanced Data Structures and Algorithms

9. CS223 Advanced Data Structures and Algorithms
Insert
CS223 Advanced Data Structures and Algorithms

10. CS223 Advanced Data Structures and Algorithms
Insert
CS223 Advanced Data Structures and Algorithms

11. CS223 Advanced Data Structures and Algorithms
Insert
CS223 Advanced Data Structures and Algorithms

12. CS223 Advanced Data Structures and Algorithms
Delete 15 15 5 3 10 6 7 23 16 18 20 12 5 16
Case 1: 3 12 20 18 23 10 13 6 7 15 5 3 10 6 7 23 16 18 20 13 12 15 5 3 10 6 7 23 18 20 13 12
Case 2: 15 5 3 10 6 7 23 16 18 20 13 12 15 5 3 6 23 16 18 20 13 12 10 7
Case 3:
CS223 Advanced Data Structures and Algorithms

13. CS223 Advanced Data Structures and Algorithms
Delete
In terms of deletion, everything boils down to delete a leaf.
Deletion of a red leaf is trivial.
How to delete a black leaf: Change the color of the leaf to be deleted to red by a top-down pass.
CS223 Advanced Data Structures and Algorithms

14. Three Cases When X Has Two Black Children
CS223 Advanced Data Structures and Algorithms