AVL Search Tree
Presented By:
Prof. U V Thete
Dept of Computer Sci.
YMA
AVL TREES:
Defination of AVL trees
Why AVL Trees?
Properties of AVL trees
Construct of AVL trees
Insertion and Deletion
DEFINATION OF AVL TREES

An AVL tree is another balanced binary search tree
 AVL Trees
=Adelson- Velskii and L andis trees comes from
its inventors described in 1962 for the organization of
information.“
 Example:
12
8
5
4
18
11
17
WHY AVL TREES?

When sorted elements inserted into BST than it becomes as
degenarated tree
12

Which is insufficient

After insertion/deletion its height will be increase

And also can be imbalanced
13
18
That is ensure AVL trees!
20
AVL TREES:
12
12
8
5
18
11
17
13
18
20
4
PROPERTIES OF AVL TREES:



Sub-trees of each node can differ by at most
1 in their height
So, h=max(l.h-r.h)+1
h=Height
l=left, r=Right
Calculate the height
By default, leaf height =0.
 And empty sub-trees have a Height of -1.

Height = max(L.height, R.height) + 1
sub-tree L has a
height of 0
55
Height 0 32
Height 0 64
Height 2
71
sub-tree R has a
height of 1
Height 1
86
Height = 0 = max(-1,-1) + 1
AVL (ADELSON-VELSKII AND LANDIS)
TREES
44
4
9
• An AVL Tree is a
binary search tree
such that for every
internal node v of T,
the heights of the
children of v can differ
by at most 1.
2
17
78
1
3
2
32
88
50
1
48
62
1
An example of an AVL tree where the
heights are shown next to the nodes:
1
AVL (ADELSON-VELSKII AND LANDIS)
TREES

AVL tree is a binary search tree with balance
condition

Balance condition

For every node in the tree, height of left and right subtree
can differ by at most 1
10
To ensure depth of the tree is O(log(N))
 And consequently, search/insert/remove complexity bound
O(log(N))

AVL TREES:
12
12
8
8
18
5
5
11
17
4
4
11
7
2
YES
H=3-2
=1
18
NO
H=4-2
=2
17
ROTATIONS:
Rotations are to rearrange nodes to maintain
AVLness.
 They are needed on insert and delete.

NAME OF ROTATIONS

There are four of them
Single Left (LL) and Single Right (RR)
 Double Left (LR) and Double Right (RL)

The names describe which way the node moves. For
example, a left rotation moves the node down and left.
 Nodes always move down when rotated.

Double Rotation
A LR double rotation is to rotate something to the left,
and then its former parent to the right.
 A RL double rotation is to rotate something to the right,
and then its former parent to the left.

Insertion and Deletions
 It
is performed as in binary search trees
 After
insertion or del if it will be imbalanced, Than
we need rotation to correct balance

Single Rotation

Double Rotation, Both can be Left/Right rotation.
 For
insertions, one rotation is sufficient
 For
deletions, may needed double
AVL TREE INSERT AND REMOVE



Do something to fix it : rotations
After rotations, the balance of the whole tree
is maintained
16

Do binary search tree insert and remove
The balance condition can be violated
sometimes
BALANCE CONDITION VIOLATION

If condition violated after a node insertion


Rebalance the tree through rotation at the deepest node
with balance violated


The entire tree will be rebalanced
Violation cases at node k (deepest node)
An insertion into left subtree of left child of k
2. An insertion into right subtree of left child of k
3. An insertion into left subtree of right child of k
4. An insertion into right subtree of right child of k
1.

Cases 1 and 4 equivalent


Single rotation to rebalance
Cases 2 and 3 equivalent

Double rotation to rebalance
17

Which nodes do we need to rotate?
Only nodes on path from insertion point to root may have
their balance altered
AVL TREES COMPLEXITY

Overhead
Extra space for maintaining height information at
each node
 Insertion and deletion become more complicated, but
still O(log N)

Advantage

Worst case O(log(N)) for insert, delete, and search
18

Construct of AVL Trees:
Now we build an AVL tree: 15, 20, 24, 10, 13, 7, 30, 36,
25
20
15
20
15
24
10
13
24
15, 20, 24, 10, 13, 7, 30, 36, 25
20
20
15
13
10
24
13
10
24
15
15, 20, 24, 10, 13, 7, 30, 36, 25
20
13
10
7
13
10
24
15
7
20
15
24
30
36
15, 20, 24, 10, 13, 7, 30, 36, 25
13
13
10
10
20
7
7
20
15
15
30
30
24
24
36
36
25
15, 20, 24, 10, 13, 7, 30, 36, 25
13
10
7
13
20
15
10
24
7
30
25
24
20
15
36
30
25
36
Remove Operation in AVL Tree
Removing a node from an AVL Tree is the same as
removing from a binary search tree. However, it may
unbalance the tree.
 Similar to insertion, starting from the removed node we
check all the nodes in the path up to the root for the first
unbalance node.
 Use the appropriate single or double rotation to balance the
tree.
 May need to continue searching for unbalanced nodes all
the way to the root.

Deletion
Remove 24 from the AVL tree.
13
10
13
10
24
7
20
15
7
30
25
36
20
15
30
25
36
Deletion
Remove 20 from the AVL tree.
13
10
7
13
20
15
10
30
30
7
25
15
36
36
25
Deletion
After Remove 24 and 20 from the AVL tree.
13
13
10
7
30
15
10
36
25
Fig (a):Unbalanced
15
7
30
25
Fig(b): Balanced
36
Thank you
All