Left Rotations

1
Left Rotations
Right subtree is deeper than the left subtree
so that the balance factor at node X becomes
2. (Node X is the node about which the rotation
will occur.)
2
X

• AVL tree unbalanced at node 46.
• The 2 balance factor indicates need for a
  left rotation.

3
X
2. (a) Child node 65 interchanges with parent
46. (b) Resulting tree is not a valid search
tree.
4
24
0
10
65
1
0
15
46
80
0
0
0
3. (a) Node 46 must become a left child of 65.
(b) Node 80 moves up one level in right subtree
of 65. (c) Balance factors are recomputed.
AVL tree is balanced.
5
Right Rotations
Left subtree is deeper than the right subtree
so that the balance factor at node X becomes
-2. (Node X is the node about which the rotation
will occur.)
6
X

• Tree is unbalanced at node containing 46.
• Balance factor of 2 indicates need for a
  right rotation.

7
X
2. (a) Node 35 interchanges with parent node 46.
(b) Resulting tree is not a valid search tree.
8
54
0
80
35
-1
0
65
46
20
0
0
0
3. (a) Node 46 must become a right child of node
35. (b) Node 20 remains left child of 35 but
moves up 1 level. (c) Balance factors are
recomputed. AVL tree is balanced.
9
Double Rotations

• Insertion occurs in either
• In the right subtree of the left child of node X.
• or
• (2) In the left subtree of the right child of
  node X.
• (Node X is the node which becomes unbalanced)

10

• Tree is unbalanced at node containing 8 due to
  insertion
• of node 5 into right subtree of left child
  of node 8.

11
2. (a) Rotate Xs grandchild about Xs child
(rotate 6 about 4). (b) Resulting tree is not
AVL balanced.
12
3. (a) Rotate 6 about 8. (Rotate Xs new child
about X.) (b) Tree becomes AVL balanced.