A red-black tree is a type of self balancing binary search tree, with one extra attribute for each node: the colour, which is either red or black. A red–black tree is a binary search tree that inserts and deletes in such a way that the tree is always reasonably balanced.

1. PRAVIN D’SILVA
1109
MCA , GOA UNIVERSITY

2. Acknowledgement
I would like express my sincere gratitude to Dr.
Jyoti Pawar, for the support and guidance as
well as the feedback she has provided
throughout this study.

3. Definition
• A red-black tree is a type of self balancing
binary search tree, with one extra attribute for
each node: the colour, which is either red or
black.
• A red–black tree is a binary search tree that
inserts and deletes in such a way that the tree is
always reasonably balanced.

4. Balanced binary tree
• A non-empty binary tree T is
balanced if:
1) Left subtree of T is balanced
2) Right subtree of T is balanced
3) The difference between heights of
left subtree and right subtree is not
more than 1.

5. Red-Black tree
• Recall binary search tree
▫ Key values in the left subtree <= the node value
▫ Key values in the right subtree >= the node value
• Operations:
▫ insertion, deletion
▫ Search, maximum, minimum, successor, predecessor.
▫ O(h), h is the height of the tree.

6. Red-black trees
•
Definition: a binary tree, satisfying:
1.
2.
3.
4.
5.
•
•
Every node is red or black
The root is black
Every leaf is NIL and is black
If a node is red, then both its children are black
For each node, all paths from the node to
descendant leaves contain the same number of
black nodes.
Purpose: keep the tree balanced.
Other balanced search tree:
▫
AVL tree, 2-3-4 tree, Splay tree, Treap

7. Example of red black trees
Each Red Node can have only Black children

8. Not a red black tree

9. Black height of a red black tree
• Black height does not count the root itself.
• we use "nil leaves" or "null leaves", which contain no data and
merely serve to indicate where the tree ends
7
BH=2
BH=2
3
BH=1
BH=0
22
BH=1
8
10
BH=1,
ignore red
nodes!!
NIL
pointer

10. Complexity
• The persistent version of red-black trees
requires O(log n) space for each insertion or
deletion.

11. Some operations in log(n)
• Search, minimum, maximum,
successor, predecessor.
• Let us discuss insert and delete.

12. Inserting in a red black tree
• Let k be the key being inserted.
• As in case of a BST, we first search for k; this
gives us the place where we have to insert k.
• We create a new node with key k and insert it at
this place
• The new node is colored red.

13. • Since inserted node is Red , the black height of
the tree remains unchanged.
• However if the parent node is also red, then we
have a double red problem
k
k
NO PROBLEM
DOUBLE RED PROBLEM

14. INSERTION : CASE 1
• Parent of node (a) must be black (b).
• The other child of (b) is black (c) .
• Rotation corrects the defect.
b
c
a
b
a
k
c
k

15. Insertion case 2
• Parent of node (a) is red
• Parent of (a) must be black (b)
• The other child of (b) is also red (c)
c|b|a
b
b
h+1
c
h
c
a
k
h
k
h
h
h
h+1
a
h
h
h
h
h

16. Deletion
• To delete a node we proceed as in a BST
• Hence the node which is deleted is the parent of
an external node
• Hence it is either a leaf or parent of a leaf
• Steps:
▫
▫
▫
▫
Search
Identify
Leaf, then delete
If internal node, find successor or predecessor,
swap, then delete the successor or predecessor

17. Consider a red black
a
a
b
c
b
c

18. Case 1
a
c
b
a
h-1
h-1
h-1
h-1
h-1
h-1
h-1
h-1
b
a
a
b
h-1
h-1 h-1
h-1
c
|
h-1
h-1
h-1
h-1
| b| c
| a

19. Case 2
•
•
•
•
If parent is a red node (a)
Then it has a child (b) which must be black
If (b) has no red child
Recoloring solves the problem
a
a
b
b
h-1
h-1
h-1
h-1
h-1
h-1

20. Case 3
a
a
b
b
h-1
h-1
h-1
h-1
h-1
h-1

21. Deletion Summary
• In most cases, deletion can be completed by
simple rotation/ coloring.
• In case 3, the height of the subtree reduces and
so we need to proceed up the tree
• But in case 3, we only recolor the nodes
• Thus, if we proceed up the tree, then we only
need to recolor. Eventually we would do a
rotation.

22. References
• "Introduction to Algorithms, Third Edition," by Thomas H.
Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein.
• Lecture Series on Data Structures and Algorithms by Dr. Naveen
Garg, Department of Computer Science and Engineering, IIT Delhi.
http://nptel.iitm.ac.in
• Lecture 10: Red-black Trees, Rotations, Insertions, Deletions:
http://videolectures.net/mit6046jf05_demaine_lec10/
• http://mitpress.mit.edu/algorithms/solutions/chap13-solutions.pdf
• http://www.cs.purdue.edu/homes/ayg/CS251/slides/chap13c.pdf
• Left leaning Red black trees
• http://www.cs.princeton.edu/~rs/talks/LLRB/RedBlack.pdf
• http://www.cs.princeton.edu/~rs/talks/LLRB/LLRB.pdf

23. Thank
You