CS660: AVL Trees

CS660 Combinatorial Algorithms
Fall Semester, 1996
AVL Trees

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San Diego State University -- This page last updated Oct 17, 1996

Contents of AVL Trees

References

Data Structure Techniques, Standish, 1980, section 3.7.3

Handbook of Algorithms and Data Structures, Gonnet, 1984, section 3.4.1

AVL Trees Slide # 1

Height of a tree is the length of the longest path from the root to a leaf.

A binary search tree T is a height-balanced k-tree or HB[k]-tree, if each node in the tree has the HB[k] property

A node has the HB[k] property if the height of the left and right subtrees of the node differ in height by at most k.
Some HB[2] Trees

AVL Trees Slide # 2

AVL Trees

A HB[1] tree is called an AVL tree.
Adel'son-Vel'skii and Landis first defined HB[1] trees

Worst Case AVL Trees

Let = the number of nodes in worst case AVL tree of height n

We have:

AVL Trees Slide # 3

Fibonacci Numbers

n
0	1	1
1	2	1
2	4	2
3	7	3
4	12	5
5	20	8
6	33	13

We get

But where

So

AVL Trees Slide # 4

Now take any AVL tree of height h with M nodes, we have

M >=

So M + 2 >

h < 1..4404 lg(M + 2) - 0.328

Theorem ( Adel'son-Vel'skii and Landis 1962) The height of a balanced tree with N internal nodes always lies between lg (N +1 ) and 1..4404 lg(M + 2) - 0.328

AVL Trees Slide # 5

Insertion into a AVL Tree

Some notationBalance Information

Height of Subtrees

Insert key as normal binary search tree
Restructure the tree if some node are unbalanced

AVL Trees Slide # 6

Case 0Inserting into Short Subtree Balance Information ChangesNo Restructure of Tree

General Case

AVL Trees Slide # 7

Case 1a Insert into taller subtreeTaller subtree balancedTaller subtree on RightInsert into outer subsubtree Example

Perform Single Rotation

Tree is balanced

AVL Trees Slide # 8

Need for Generalized Case 1a
AVL Trees Slide # 9
Case 1a Insert into taller SubtreeTaller Subtree balancedTaller subtree on RightInsert into outer subsubtree Add node to subtree 3

Perform Single Rotation

Tree is balanced

AVL Trees Slide # 10

Case 1b Insert into taller subtreeTaller subtree balancedTaller subtree on LeftInsert into outer subsubtree This is symmetric to case 1a Example

AVL Trees Slide # 11

Case 2a Insert into taller subtreeTaller subtree balancedTaller subtree on RightInsert into inner subsubtree

Double Rotation

AVL Trees Slide # 12

Case 2a Insert into taller subtreeTaller subtree balancedTaller subtree on RightInsert into inner subsubtree Insert into subtree 2 or 3

Double Rotation

AVL Trees Slide # 13

Case 2b Insert into taller subtreeTaller subtree balancedTaller subtree on LeftInsert into inner subsubtree This is symmetric to case 2a

AVL Trees Slide # 14

Case 3a Insert into taller subtreeTaller subtree unbalancedTaller subtree on RightTaller subsubtree on RightInsert into outer subsubsubtree Insert K

Reduces to Case 1aInsert K

We have covered all cases!

AVL Trees Slide # 15

Deletions from an AVL Tree

Delete the node as in a binary search tree

The node deleted will be either a leaf or have just one subtree

Since this is an AVL tree if the deleted node has one subtree, then that subtree contains only one node

Traverse up the tree from the deleted node checking the balance of each node

AVL Trees Slide # 16

Case 1a Traversing Up from Left subtreeDeletion reduced the height of the left subtreeNode has equal balance
Action:

Change balance of the node and stop
Have not effected the balance of any higher nodes

Case 1b Traversing Up from Right subtreeDeletion reduced the height of the Right subtreeNode has equal balance
Action:

Same as 1a

AVL Trees Slide # 17

Case 2a Traversing Up from Left subtreeDeletion reduced the height of the left subtreeLeft subtree was greater then the right subtree
Action:

Change balance of the node and stop
May have effected the balance of any higher nodes, so continue up tree

Case 2b Traversing Up from Right subtreeDeletion reduced the height of the Right subtreeRight subtree was greater then the Left subtree
Action:

Similar to 2a

AVL Trees Slide # 18

Case 3a Traversing Up from Left subtreeDeletion reduced the height of the left subtreeRight subtree was greater then the Left subtree Left subtrees has equal subsubtrees
Action:

Perform single rotation, Adjust the balance
Have not effected the balance of any higher nodes
Stop here

Single Rotation

AVL Trees Slide # 19

Case 4a Traversing Up from Left subtreeDeletion reduced the height of the left subtreeRight subtree was greater then the Left subtree One or both of B subtrees has height h-1
Action:

Double rotation at B
May have effected the balance of any higher nodes, so continue up tree

Double Rotation

AVL Trees Slide # 20

Case 5a See picture
Action:

Single rotation at B
May have effected the balance of any higher nodes, so continue up tree

AVL Trees Slide # 21

Performance of AVL Trees

Theorem. An insertion into an AVL tree requires at most one rotation to rebalance a tree. A deletion may require lg(N) rotations to rebalance the tree.
Simulation Results AVL

n	C(n)	E[ h(n) ]	R(n)
5	2.2	3.0	0.213
10	2.907	4.0	0.318
50	4.930	6.947	0.427
100	5.889	7.999	0.444
500	8.192	10.923	0.461
1000	9.202	11.998	0.463
5000	11.555	14.936	0.465
10000	12.568	15.996	0.465

n = number of nodes in the tree
C(n) = average number of comparisons to find a key
E[ h(n) ] = expected height of tree
R(n) = average number of rotations per insertion/deletion

CS660 Combinatorial Algorithms Fall Semester, 1996 AVL Trees