This document discusses height balanced binary trees. It begins by defining a height balanced binary tree as one where the depth of the two subtrees of every node never differs by more than 1. It then lists different types of binary trees including height balanced trees. The rest of the document discusses properties of height balanced trees such as searching, insertion and deletion operations. It provides examples of rotations performed during insertion to maintain the height balanced property, covering left-left, right-right, left-right and right-left cases. Finally, it includes pseudocode for calculating the height of a tree.

1. HEIGHT BALANCED
TREE
M.NIVETHITHA,
DEPARTMENT OF INFORMATION TECHNOLOGY,
V.V.VANNIAPERUMAL COLLEGE FOR WOMEN,
VIRUDHUNAGAR.

2. BINARY TREE
2
a binary tree is a tree data structure in which each node has at most
two children, which are referred to as the left child and the right child.

3. Types of binary trees
1. Height balanced tree
2.Binary search tree
3.Heap tree
4.Threaded binary tree
5.Expression tree
3

4. 1.Height balanced tree

5. “Height-balanced binary
tree : is defined as a
binary tree in which the
depth of the two sub trees
of every node never differ
by more than 1.”
5

6. definition
> A binary search tree is said to be a height
balanced binary search tree if all its nodes have
a balance factor of
1,0 or -1.
That is,
|bf|=|hL – hR| ≤ 1
For every node in tree.
6

7. example
7

8. The basic objectives of the height balanced tree
8
1.Searching
2.Insertion
3.deletion
These operations may not be with the minimum time but the time
involved is less than that of in an unbalanced binary search
tree.

9. 9
How an unbalanced
binary tree converted
into a height balanced
tree
Suppose initially there
is a height balanced tree

10. 10
The mechanism to balance an unbalance tree due to the insertion of a node
into it.
Steps :
1.Insert node into a binary search tree :
insert the node into its proper position following the properties of
binary tree
2.Compute the balance factors:
on the path starting from the root node to the node newly inserted,
compute the balance factors of each node.
3.Decide the pivot node :
on the path as traced in step 2, determine whether the absolute
value of any nodes balance factor is switched from 1 to 2.if so, the tree

11. 11
becomes unbalanced. It is called Pivot node.
4.Balance the unbalance tree :
to manipulate pointers centred at the pivot node to bring the tree back
into height balance. This pointer manipulation is known as AVL Rotation
AVL ROTATIONS:
An elegant method was devised in 1962
By two Russian mathematicians ,
G.M.Adelson-Velskii
E.M.Landis

12. 12
There are four cases of rotations
CASE 1 : LEFT TO LEFT INSERTION
 RIGHT SUB TREE OF LEFT CHILD OF PIVOT NODE BECOMES THE LEFT SUB TREE OF P.
 P BECOMES THE RIGHT CHILD OF A
 LEFT SUB TREE OF A REMAINS THE SAME
A
P
PR
AR
AL
A
P
AR
AL
PR

13. ALGORITHM
13
INPUT : Pointer Pptr to the pivot node
OUTPUT : AVL rotations corresponding to the unbalance due to insertion in the left sub tree
of the left child of Pptr
STEPS :
1.Aptr = Pptr - > LCHILD
2.Pptr-> LCHILD = Aptr->RCHILD
3.Aptr->RCHILD=Pptr
4.Pptr->HEIGHT=Compute Height(Pptr)
5.Aptr->HEIGHT=Compute Height(Aptr)
6.Pptr=Aptr
7.Stop

14. CASE 2 : RIGHT TO RIGHT INSERTION
14
 LEFT SUB TREE OF RIGHT CHILD OF PIVOT NODE BECOMES THE RIGHT SUB TREE OF P.
 P BECOMES THE LEFT CHILD OF A
 RIGHT SUB TREE OF A REMAINS THE SAME
A
P
AR
AL
PL
A
P

15. ALGORITHM
15
INPUT : Pointer Pptr to the pivot node
OUTPUT : AVL rotations corresponding to the unbalance due to insertion in the right sub
tree of the right child of Pptr
STEPS :
1.Aptr = Pptr - > RCHILD
2.Pptr-> RCHILD = Aptr->LCHILD
3.Aptr->LCHILD=Pptr
4.Pptr->HEIGHT=Compute Height(Pptr)
5.Aptr->HEIGHT=Compute Height(Aptr)
6.Pptr=Aptr
7.Stop

16. CASE 3 : LEFT TO RIGHT INSERTION
16
ROTATION 1 :
 LEFT SUB TREE OF THE RIGHT CHILD OF THE LEFT CHILD OF PIVOT NODE BECOMES
THE RIGHT SUB TREE OF THE LEFT CHILD.
 LEFT CHILD OF THE PIVOT NODE BECOMES THE LEFT CHILD OF B
ROTATION 2 :
 RIGHT SUBTREE OF THE RIGHT CHILD OF THE LEFT CHILD OF THE PIVOT NODE
BECOMES THE LEFT SUBTREE OF P
 P BECOMES THE RIGHT CHILD OF B
P
PR
AL
A
B
BL
BR
B
A
AL
BL
P
BR PR

17. ALGORITHM
17
INPUT : Pointer Pptr to the pivot node
OUTPUT : AVL rotations corresponding to the unbalance due to insertion in the right sub
tree of the left child of Pptr
STEPS :
1.Aptr = Pptr - > LCHILD
2.RightToRightRotation(Aptr)
3.LeftToLeftRotation(Pptr)
4.Stop

18. CASE 4 : RIGHT TO LEFT INSERTION
18
ROTATION 1 :
 RIGHT SUB TREE OF THE LEFT CHILD OF THE RIGHTCHILD OF PIVOT NODE BECOMES
THE LEFT SUB TREE OF A.
 RIGHT CHILD OF THE PIVOT NODE BECOMES THE RIGHT CHILD OF B.
ROTATION 2 :
 LEFT SUBTREE OF THE RIGHT CHILD OF THE RIGHT CHILD OF THE PIVOT NODE
BECOMES THE RIGHT SUBTREE OF P
 P BECOMES THE LEFT CHILD OF B
P
A
AR
PL
B
BL
BR
B
P
PL
BL
A
BR AR

19. ALGORITHM
19
INPUT : Pointer Pptr to the pivot node
OUTPUT : AVL rotations corresponding to the unbalance due to insertion in the left sub tree
of the right child of Pptr
STEPS :
1.Aptr = Pptr - > RCHILD
2. LeftToLeftRotation(Aptr)
3.RightToRightRotation(Pptr)
4.Stop

20. IMPLEMENTATION FOR
HEIGHT BALANCING
TREE
LCHILD DATA HEIGHT RCHILD

21. ALGORITHM
21
INPUT : Pointer PTR to the tree whose height has to be calculated
OUTPUT : Height of the tree
DATA STRUCTURE : Linked Structure of Height
STEPS :
1.If (PTR=NULL) then
2. Height = 0
3. Return(height)
4.Else
5. lptr=PTR->LCHILD
6. Rptr=PTR->RCHILD
7. Hl=Compute Height(lptr)
8. Hr=Compute Height(Rptr)
9. If(Hl <= Hr) then
10. height = 1 + Hr
11. Else
12. height = 1 + Hl
13. Endif
14. Return(height)
15. Endif
16. stop

22. 22