A red-black tree is a binary search tree with one extra bit of storage per node: its color, which can be either RED or BLACK.

1. Red-Black Trees
• Name : TCHAYE-KONDI JUDE
Simulator Link : https://www.cs.usfca.edu/~galles/visualization/RedBlack.html

2. RESUME
I. INTRODUCTION
1. TREE
2. BINARY TREE
3. BINARY SEARCH TREE
II. RED BLACK TREE
III. OPERATIONS

3. INTRODUCTION

4. Tree
• Tree is a non-linear data structure which organizes data in hierarchical
structure and this is a recursive definition.

5. • Root • Edge

6. • Leaf

7. • Internal Nodes

8. • Level

9. • Height

10. • Path

11. BINARY TREE
• A tree in which every node can have a maximum of two children is called as Binary
Tree.
• A binary tree has the following time complexities :
• Search Operation - O(n)
• Insertion Operation - O(n)
• Deletion Operation - O(n)

12. BINARY SEARCH TREE
• Binary Search Tree is a binary tree in
which every node contains only smaller
values in its left subtree and only larger
values in its right subtree.

13. BINARY SEARCH TREE
• left subtree of every
node contains nodes
with smaller values
and right subtree of
every node contains
larger values.
Algorithm Average Worst case
Search O(log n) O(n)
Insert O(log n) O(n)
Delete O(log n) O(n)
Time complexity

14. Balanced Binary Search Trees
• A Binary Search Tree (BST) of N nodes is balanced if height is in O(log N)
• A balanced tree supports efficient operations,
• There are many implementations of balanced BSTs, including AVL trees,
RedBlack trees and AA trees.

15. RED BLACK TREE
• A red-black tree is a binary search tree with one extra bit of storage per node: its
color, which can be either RED or BLACK.
• By constraining the node colors on any simple path from the root to a leaf, red-
black trees ensure that no such path is more than twice as long as any other.
• Each node of the tree now contains the attributes color, key, left, right, and p.
• If a child or the parent of a node does not exist, the corresponding pointer attribute
of the node contains the value NIL.

17. PROPERTIES
• A red-black tree is a binary tree that satisfies the following red-black
properties:
1. Every node is either red or black.
2. The root is black.
3. Every leaf (NIL) is black.
4. If a node is red, then both its children are black.
5. For each node, all simple paths from the node to descendant leaves contain the same
number of black nodes.

18. BLACK-HEIGHT
• the number of black nodes on any simple path from, but not including, a
node x down to a leaf the black-height of the node, denoted bh(x) or B
• We define the black-height of a red-black tree to be the black-height of its
root.

19. OPERATIONS
&
COMPLEXITY

20. OPERATIONS
1. INSERTION, DELETION, ROTATIONS
2. SEARCH, MINIMUM, MAXIMUM, SUCCESSOR, AND PREDECESSOR…

21. THE NUMBER OF INTERNAL
NODES OF A NODE X
N(X) ≤ 𝟐 𝒃𝒉(𝒙) − 𝟏

23. INTERNAL NODES OF X
Let prove by induction that : the subtree rooted at any node x contains at least 𝟐 𝒃𝒉(𝒙)
− 𝟏
internal nodes.
1. If the height of x is 0 then x must be a leaf 𝟐 𝒃𝒉(𝒙) − 𝟏 = 𝟐 𝟎 − 𝟏 =
0
2. For the inductive step, consider a node x that has positive height and is an internal node with
two children :
• Each child has a black-height of either bh(x) or bh(x) - 1, depending on whether its color is
red or black, respectively.
• We conclude that each child has at least : 𝟐 𝒃𝒉(𝒙)−𝟏
− 𝟏 internal nodes (the height of a child
of x is less than the height of x itself)
3. each child has at least 𝟐 𝒃𝒉(𝒙)−𝟏
− 𝟏 internal nodes.
Then the subtree rooted at x contains at least (𝟐 𝒃𝒉(𝒙)−𝟏 − 𝟏)+(𝟐 𝒃𝒉(𝒙)−𝟏 − 𝟏) +1 = 𝟐 𝒃𝒉(𝒙) − 𝟏

24. Conclusion
• THE NUMBER OF INTERNAL NODES OF A NODE X ( N(X) )
• N(X) ≤ 𝟐 𝒃𝒉(𝒙)
− 𝟏

25. HEIGHT OF A RB
TREE
Knowing the number of internal nodes (n)

26. HEIGHT OF A RB TREE(H)
• A red-black tree with n internal nodes has height at most H ≤ 2lg(n+ 1).
According to property 4(If a node is red, then both its children are black.), at least
half the nodes on any simple path from the root to a leaf, not including the
root, must be black. Consequently, the black-height of the root must be at
least h=2; thus,
As an immediate consequence of this lemma, we can implement the dynamic-set
operations SEARCH, MINIMUM, MAXIMUM, SUCCESSOR, and
PREDECESSOR in Olg(n) time on red-black trees,

27. OPERATIONS
They do not directly support the dynamic-set operations INSERT and DELETE,
since they do not guarantee that the modified binary search tree will be a red-black
tree.

29. ROTATIONS

33. LEFT ROTATION :

34. RIGHT ROTATION

35. Rotation is followed by Recolor operation

37. INSERTION

38. INSERTION
• We can insert a node into an n-node red-black tree in Olg(n) time.

39. INSERTION

40. Big-O Complexity
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