Submitted to:

Miss.Komal Bashir

Submitted by:

Maira shahnawaz

Uneza rehman

Amna siddiqui

Binary Search Tree (BST) is a good

data structure and It supports:

Insertion

Deletion

Search

Red-Black Tree is one of the balanced binary search

tree

A red-black tree is a binary search tree with one

extra attribute for each node: the colour, which is

either red or black

Invented 1972

Invented by Rudolf Bayer

Time complexity in big (o) notation:

Average Worst case

Space O(n) O(n)

Search O(log n) O(log n)

Insert O(log n) O(log n)

Delete O(log n) O(log n)

That's really your decision, but I've found that red black

trees are best suited to largely random data that has

occasional degenerate runs, and searches have no

locality of reference. This takes full advantage of the

minimal work that red black trees perform to maintain

balance compared to AVL trees and still allows for

speedy searches.

A binary search tree is a red-black tree if:

Every node is either red or black

The root is black

Every leaf (NIIL) is black

A red node’s children are black

Every path from a node x to an external node must

contain the same number of black nodes = black-

height(x)

Red Black Tree

Since red-black tree is a balanced BST, it supports

Search

Insertion

Deletion

A new item is always inserted as a leaf in the tree

If we color a leaf black, we will create a longer path

of black nodes (violating property 4)

Therefore, a new item must be colored red [unless it

is the root]

if the parent is colored black, we are done

If the parent is red, we will have consecutive red

nodes (violating property 3)

We must adjust the tree to ensure property three,

without introducing a violation to property 4

The operations are rotations and color changes.

Sibling of the parent is black [adopt the convention

that null references are black]

Inserted node is an outside grandchild

A single rotation between the parent and the

grandparent, with appropriate color changes,

restores property 3

Notice that before insertion of node X, there was

one black node from G to each of A, B, and C, and

two black nodes from G to each of D and E

After the rotation and recoloring, notice that the

number of black nodes on each of those paths

remains unchanged

Property 3 has been restored

Sibling of the parent is red

Neither single nor double rotations work, since both

result in (possibly) consecutive red nodes

Case 2

This fixes property 3 for this sub tree

What happens if the parent of this sub tree is also

red?

We could percolate this procedure up toward the

root until we no longer have two consecutive re

nodes, or we reach the root

The advantage over AVL trees has disappeared

Color flip

Single Rotation

Recall that in deleting from a binary search tree, the

only nodes which are actually removed are leaves or

nodes with exactly one child

Nodes with two children are never removed. Their

contents are just replaced

Deletions

If the node to be deleted is red, there is no problem -

- just delete the node

If the node to be deleted is black, its removal will

violate property 4

The solution is to ensure that any node to be deleted

is red

Remove 9

Remove 8:

Remove 7:

Although you may never need to implement your own

set or map classes, thanks to their common built-in

support, understanding how these data structures work

should help you better assess the performance of your

applications and give you more insight into what

structure is right for a given task. For more practice with

these concepts, check out these problems from the Top

Coder archive that involve trees: