Download Presentation
## Left Rotations

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**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.)**24**+1 10 +1 46 +2 15 0 65 +1 80 0 X • AVL tree unbalanced at node 46. • The +2 balance factor indicates need for a left rotation.**24**+1 10 +1 65 +2 15 0 46 +1 80 0 X 2. (a) Child node 65 interchanges with parent 46. (b) Resulting tree is not a valid search tree.**24**0 10 +1 65 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.**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.)**54**-1 80 -1 46 -2 65 0 35 -1 20 0 X • Tree is unbalanced at node containing 46. • Balance factor of –2 indicates need for a right rotation.**54**-1 80 -1 35 -2 65 0 46 -1 20 0 X 2. (a) Node 35 interchanges with parent node 46. (b) Resulting tree is not a valid search tree.**0**54 80 35 -1 0 65 20 46 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.**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)**12**-2 8 16 -2 -1 X 4 10 +1 0 14 0 2 6 0 -1 5 0 • Tree is unbalanced at node containing 8 due to insertion • of node 5 into right subtree of left child of node 8.**12**-2 8 16 -2 -1 X 6 10 -2 0 14 0 4 0 2 5 0 0 2. (a) Rotate X’s grandchild about X’s child (rotate 6 about 4). (b) Resulting tree is not AVL balanced.**12**-1 6 16 0 -1 X 4 8 +1 0 14 0 2 5 10 0 0 0 3. (a) Rotate 6 about 8. (Rotate X’s new child about X.) (b) Tree becomes AVL balanced.