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Data Structures and Algorithms

(CS210/ESO207/ESO211)

Lecture 16

Height balanced binary search tree

Red-black trees (Description and handling insertion)

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A rooted binary tree

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root

edgesNodes

leaves

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Terminologies

Fields associated with a node in T:

1. value

2. left(pointer to left child)

3. right(pointer to right child)

4. parent(pointer to parent)

Height(T): The maximum number of nodes on any path from root to a leaf node.

(Note that this definition of height is a bit different from the one we gave in lecture 8.)

Depth(v): The number of nodes on the path from root to node vin T. Depth of a node

is also called its level. For example, depth/level of root of T is 1, depth/level of a child

of root is 2, and so on.

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Terminologies

Full binary tree:

A binary tree where every internal node has exactly two children.

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This node has exactly one child. So

the current tree is not a full binary

tree.

This is now a full

binary tree.

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Binary Search Tree

Definition:A Binary Tree Tstoring values is said to be Binary Search Tree if for each node vin T

If left(v) NULL, then value(v) > value of every node in subtree(left(v)).

If right(v)NULL, then value(v) < value of every node in subtree(right(v)).

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root

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Binary Search Tree: a slight transformation

Henceforth, for each NULLchild link of a node in a BST, we create a NULLnode.

As a result,

Each leaf node in a BST will be a NULLnode.

The BST will always be a full binary tree. 6

root

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This transformation is merely to help us in

the analysis of red-black trees. It does not

cause any extra overhead of space. All

NULL nodescorrespond to a single node

in memory.

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A fact we noticed in our previous discussion on

BSTs (Lecture 8)

Time complexity of Search(T,x) and Insert(T,x) in a Binary Search

Tree T= ??

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O(Height(T))

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Time complexity of Searching and inserting in a

perfectly balanced Binary Search Tree on nnodes

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9625

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T

O( log n) time

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Time complexity of Searching and inserting in a

skewed Binary Search tree on nnodes

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T2

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O(n) time !!

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A height balanced BST

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Definition: A BST storing nelements is said to be heightbalanced if its height is O(log n).

Examples: AVL Trees

Red Black Trees

Splay trees

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Red Black Tree

A height balanced BST

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RedBlackTree

Red-Black tree is a binary search tree satisfying the following properties at

each stage.

Each node is colored redor black.

Each leaf is colored black and so is the root.

Every red node will have both its children black.

The number of blacknodes on a path from root to a leaf node is same for

every leaf. This parameter is called blackheight of the tree.

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A binary search tree

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head

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Can you color the

nodes to make it a

red-black tree ?

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A binary search tree

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head

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A Red Black Tree

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Why is a red black tree height balanced ?

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This could not be explained properly in this class. So

it will be discussed again in the next lecture (12

September).

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Insertion in a red-black tree

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T

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Let us insert 83into T.

What color should

we assign to the

new node ?

We hall assign every

newly inserted node a

redcolor. (give reason)

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Insertion in a red-black tree

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T

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Let us insert 54into T.

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Color imbalance

We have again a red

black tree.In order to remove the color

imbalance try flipping the colors

of parent (and uncle) of the new

node with the grandparent

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Insertion in a red-black tree

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T

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Insertion in a red-black tree

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T

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Let us insert 44into T.

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Color imbalance

In order to remove the color

imbalance try flipping the colors

of parent (and uncle) of the

new node with the grandparent

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Insertion in a red-black tree

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T

2

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Color imbalance

Do the same

trick again.

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Insertion in a red-black tree

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T

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Color imbalance is removed. But the

root is rednow.

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Insertion in a red-black tree

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T

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Insertion in a red-black treesummary till now

Letpbe the newly inserted node. Assign redcolor top.

Case 1:parent(p) is black

nothing needs to be done.

Case 2:parent(p) is red and uncle(p) is red,

Swap colors ofparent(and uncle) with grandparent(p).

This balances the color atpbut may lead to imbalance of color at

grandparent of p. Sopgrandparent(p), and proceed upwards similarly.

If in this mannerp becomes root, then we color it black.

Case 3:parent(p) is red and uncle(p) is black.

This is a nontrivial case. So we need some more tools .

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Rotation around a node

An important tool for balancing trees

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Rotationaround a node

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v

u

xRight rotation Left rotation

v

u

x

p

Note that the tree T continues to

remain a BST even after rotation

around any node.

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Handling case 3

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Description of Case 3

pis a redcolored node.

parent(p) is also red.

uncle(p) isblack.

Without loss of generality assume: parent(p) isleft child ofgrandparent(p).

(The case whenparent(p) isright child of grandparent(p) is handled similarly.)

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Handling the case 3two cases arise depending upon whether p is left/right child of its parent

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g

d

x

b

kp

g

d

x

k

f

p

Case 3.1:

p is left child of its parentCase 3.2:

p is right child of its parent

Can you

transform Case

3.2 to Case 3.1

?

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Handling the case 3two cases arise depending upon whether p is left/right child of its parent

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Case 3.2:

p is right child of its parent

g

f

x

k

d

21

3

Vow!

This is exactly Case 3.1

g

d

x

k

f

2

1

3

left rotation

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We need to handle onlycase 3.1

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Handling the case 3.1

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g

d

x

b

k

21

3f

p

Can we say

anything about

color of node f ?

It has to be

black.

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Handling the case 3.1

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g

d

x

b

k

21

3f

p

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Handling the case 3.1

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Right rotation

g

d

x

b

k

21

3f

Now every node in tree 1 has one

less blacknode on the path to root !

We must restore it. Moreover, the

color imbalance exists even now.

What to do ?

Color node d black

p g

2

d

x

b

kf1

3

p

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Handling the case 3.1

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g

d

x

b

k

21

3f

g

2

d

x

b

kf1

3

The number of blacknodes on the path to root are

restored for tree 1. Color imbalance is also removed.

But the number of blacknodes on the path to root has

increased by one for trees 2 and 3. What to do now ?

Color node g red

p

p

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Handling the case 3.1

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g

d

x

b

k

21

3f

g

2

d

x

b

kf1

3

The black height is

restored for all trees 1,2,3.

This completes Case 3.1

p

p

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Theorem: We can maintain red-black trees under insertion of

nodes in O(log n) time per insert/search operation where nis the

number of the nodes in the tree.

In the next lecture, we shall show that we can handle deletion in

a red-black tree with the same time complexity as well.

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