In the previous post, we discussed introduction to Red-Black Trees. In this post, insertion is discussed.

In AVL tree insertion, we used rotation as a tool to do balancing after insertion caused imbalance. In Red-Black tree, we use two tools to do balancing.

**1)** Recoloring

**2)** Rotation

We try recoloring first, if recoloring doesn’t work, then we go for rotation. Following is detailed algorithm. The algorithms has mainly two cases depending upon the color of uncle. If uncle is red, we do recoloring. If uncle is black, we do rotations and/or recoloring.

Color of a NULL node is considered as BLACK.

Let x be the newly inserted node.

**1)** Perform standard BST insertion and make the color of newly inserted nodes as RED.

**2)** If x is root, change color of x as BLACK (Black height of complete tree increases by 1).

**3) **Do following if color of x’s parent is not BLACK or x is not root.

….**a)**** If x’s uncle is** **RED** (Grand parent must have been black from property 4)

……..**(i)** Change color of parent and uncle as BLACK.

……..**(ii)** color of grand parent as RED.

……..**(iii)** Change x = x’s grandparent, repeat steps 2 and 3 for new x.

….**b)**** If x’s uncle is BLACK**, then there can be four configurations for x, x’s parent (**p**) and x’s grandparent (**g**) (This is similar to AVL Tree)

……..**i)** Left Left Case (p is left child of g and x is left child of p)

……..**ii)** Left Right Case (p is left child of g and x is right child of p)

……..**iii)** Right Right Case (Mirror of case i)

……..**iv)** Right Left Case (Mirror of case ii)

Following are operations to be performed in four subcases when uncle is BLACK.

## All four cases when Uncle is BLACK

**Left Left Case (See g, p and x)**

**Left Right Case (See g, p and x)**

**Right Right Case (See g, p and x)**

**Right Left Case (See g, p and x)**

**Exercise:**

Insert 2, 6 and 13 in below tree.

Please refer **C Program for Red Black Tree Insertion **for complete implementation of above algorithm.

Red-Black Tree | Set 3 (Delete)

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

## Recommended Posts:

- B-Tree | Set 2 (Insert)
- Splay Tree | Set 2 (Insert)
- K Dimensional Tree | Set 1 (Search and Insert)
- Insert a node in Binary Search Tree Iteratively
- Binary Search Tree insert with Parent Pointer
- Trie | (Insert and Search)
- 2-3 Trees | (Search and Insert)
- Treap | Set 2 (Implementation of Search, Insert and Delete)
- Efficiently design Insert, Delete and Median queries on a set
- Design a data structure that supports insert, delete, search and getRandom in constant time
- Overview of Data Structures | Set 3 (Graph, Trie, Segment Tree and Suffix Tree)
- Check if a given Binary Tree is height balanced like a Red-Black Tree
- Tournament Tree (Winner Tree) and Binary Heap
- Two Dimensional Binary Indexed Tree or Fenwick Tree
- Order statistic tree using fenwick tree (BIT)