AVL Trees

1
AVL Trees
2
Problems with Binary Search Trees

• Adding the sorted sequence A, B, C, ..., G to a
  binary search trees leads gives a tree that is no
  better than a linear list...

In fact it is less efficient than a linked list
because of the additional overhead associated
with each node.
A
3
AVL trees Basic Concepts

• Named after its two inventors, who published it
  in 1962.
• G.M. Adelson-Velskii
• E.M. Landis
• Self-balancing BST
• Balance factor the height of its right subtree
  minus the height of its left subtree
• A node with balance factor equal to -1, 0 or 1
  considered balanced
• Whenever a tree is not balanced then rotate
  around the root
• Rotate the root into the lighter subtree

4
AVL trees some properties

• The height of an AVL tree with n nodes is O(log
  n)
• The height balance property
• An AVL tree with n nodes can be searched in O(log
  n) time
• Any subtree of an AVL tree is an AVL tree
• Searching, insertion, and deletion can be done in
  O(log n) time

5
AVL trees examples

• Examples AVL trees
• Examples non-AVL trees

6
Rotation

• Definition
• To switch children and parents among two or three
  adjacent nodes to restore balance of a tree.
• A rotation may change the depth of some nodes,
  but does not change their relative ordering.
• Types
• Left rotation
• Right rotation
• Left right rotation
• Right left rotation

7
Right Rotation

• Definition
• In a binary search tree, pushing a node A down
  and to the right to balance the tree.
• Three steps (1) get the left child of node A
  node B (2) Set the right child of node B as the
  left child of node A and (3) Set node A as the
  right child of node B.

8
Left Rotation

• Definition
• In a binary search tree, pushing a node A down
  and to the left to balance the tree.
• Three steps (1) get the right child of node A
  node B (2) Set the left child of node B as the
  right child of node A and (3) Set node A as the
  left child of node B.

9
Left Right Rotation

• Definition
• Sometimes a single left rotation is not
  sufficient to balance an unbalanced tree.
• Defined to be a right rotation at the right child
  of the node followed by a left rotation at the
  node

10
Right Left Rotation

• Definition
• Similarly, sometimes a single right rotation is
  not sufficient to balance an unbalanced tree.
• Defined to be a left rotation at the left child
  of the node followed by a right rotation at the
  node

11
Rotations when and How

• To decide when you need a tree rotation is
  usually easy, but determining which type of
  rotation requires a little thought.
• General guidelines
• IF a tree is right heavy (balance factorgt1)
• IF trees right subtree is left heavy (balance
  factor lt-1)
• Perform Left Right rotation
• ELSE
• Perform single Left rotation
• IF a tree is left heavy (balance factorlt-1)
• IF trees left subtree is right heavy (balance
  factor gt1)
• Perform Right Left rotation
• ELSE
• Perform single Right rotation

12
Building an AVL tree added A
A
13
Building an AVL tree added B
A
B
14
Building an AVL tree adding C
Right hand side subtree has height of 2
Tree A unbalanced (Balance factor 2)
15
Building an AVL tree adding C
Left rotation
B
C
A
16
Building an AVL tree added D
B
C
A
D
17
Building an AVL tree adding E
B
C
A
D
E
Note Both B and C have a balance factor equal to
2. We start with the deepest node that is
unbalanced.
18
Building an AVL tree added E
It is a balanced tree
B
D
A
E
C
19
Building an AVL tree adding F
Oh, tree B is unbalanced!
B
D
A
E
C
F
20
Building an AVL tree adding F
B
D
A
E
C
F
21
Building an AVL tree added F

• Note
• D becomes Bs parent (and B becomes Ds left
  child)
• B becomes parent of Ds left child
• C becomes Bs right child

22
Building an AVL tree adding G
Oh, E becomes unbalanced!
D
E
B
F
A
C
G
23
Building an AVL tree added G
D
F
B
G
A
C
E
24
Building an AVL tree added T
D
F
B
G
A
C
E
T
Checking balance factors for every node
everything is OK!
25
Building an AVL tree adding K
D
F
B
G
A
C
E
T
Oh, G becomes unbalanced and its right subtree is
left heavy!!!
K
26
Building an AVL tree adding K
D
F
B
G
A
C
E
K
Step 1. Rotate this subtree right to make it
heavier on the right.
T
27
Building an AVL tree adding K
D
F
B
G
A
C
E
K
Step 2. Rotate this subtree left to make it to
balance it.
T
28
Building an AVL tree added K
D
F
B
K
A
C
E
T
G