1. Class No.21 Data Structures http://ecomputernotes.com
2. TreeNode<int>* singleRightRotation(TreeNode<int>* k2) { if( k2 == NULL ) return NULL; // k1 (first node in k2's left subtree) // will be the new root TreeNode<int>* k1 = k2->getLeft(); // Y moves from k1's right to k2's left k2->setLeft( k1->getRight() ); k1->setRight(k2); // reassign heights. First k2 int h = Max(height(k2->getLeft()), height(k2->getRight())); k2->setHeight( h+1 ); // k2 is now k1's right subtree h = Max( height(k1->getLeft()), k2->getHeight()); k1->setHeight( h+1 ); return k1; } k 1 k 2 Z Y X k 1 k 2 Z Y X singleRightRotation http://ecomputernotes.com
3. TreeNode<int>* singleRightRotation(TreeNode<int>* k2) { if( k2 == NULL ) return NULL; // k1 (first node in k2's left subtree) // will be the new root TreeNode<int>* k1 = k2->getLeft(); // Y moves from k1's right to k2's left k2->setLeft( k1->getRight() ); k1->setRight(k2); // reassign heights. First k2 int h = Max(height(k2->getLeft()), height(k2->getRight())); k2->setHeight( h+1 ); // k2 is now k1's right subtree h = Max( height(k1->getLeft()), k2->getHeight()); k1->setHeight( h+1 ); return k1; } k 1 k 2 Z Y X k 1 k 2 Z Y X singleRightRotation http://ecomputernotes.com
4. TreeNode<int>* singleRightRotation(TreeNode<int>* k2) { if( k2 == NULL ) return NULL; // k1 (first node in k2's left subtree) // will be the new root TreeNode<int>* k1 = k2->getLeft(); // Y moves from k1's right to k2's left k2->setLeft( k1->getRight() ); k1->setRight(k2); // reassign heights. First k2 int h = Max(height(k2->getLeft()), height(k2->getRight())); k2->setHeight( h+1 ); // k2 is now k1's right subtree h = Max( height(k1->getLeft()), k2->getHeight()); k1->setHeight( h+1 ); return k1; } k 1 k 2 Z Y X k 1 k 2 Z Y X singleRightRotation http://ecomputernotes.com
5. TreeNode<int>* singleRightRotation(TreeNode<int>* k2) { if( k2 == NULL ) return NULL; // k1 (first node in k2's left subtree) // will be the new root TreeNode<int>* k1 = k2->getLeft(); // Y moves from k1's right to k2's left k2->setLeft( k1->getRight() ); k1->setRight(k2); // reassign heights. First k2 int h = Max(height(k2->getLeft()), height(k2->getRight())); k2->setHeight( h+1 ); // k2 is now k1's right subtree h = Max( height(k1->getLeft()), k2->getHeight()); k1->setHeight( h+1 ); return k1; } k 1 k 2 Z Y X k 1 k 2 Z Y X singleRightRotation http://ecomputernotes.com
6. TreeNode<int>* singleRightRotation(TreeNode<int>* k2) { if( k2 == NULL ) return NULL; // k1 (first node in k2's left subtree) // will be the new root TreeNode<int>* k1 = k2->getLeft(); // Y moves from k1's right to k2's left k2->setLeft( k1->getRight() ); k1->setRight(k2); // reassign heights. First k2 int h = Max(height(k2->getLeft()), height(k2->getRight())); k2->setHeight( h+1 ); // k2 is now k1's right subtree h = Max( height(k1->getLeft()), k2->getHeight()); k1->setHeight( h+1 ); return k1; } k 1 k 2 Z Y X k 1 k 2 Z Y X singleRightRotation http://ecomputernotes.com
7. TreeNode<int>* singleRightRotation(TreeNode<int>* k2) { if( k2 == NULL ) return NULL; // k1 (first node in k2's left subtree) // will be the new root TreeNode<int>* k1 = k2->getLeft(); // Y moves from k1's right to k2's left k2->setLeft( k1->getRight() ); k1->setRight(k2); // reassign heights. First k2 int h = Max(height(k2->getLeft()), height(k2->getRight())); k2->setHeight( h+1 ); // k2 is now k1's right subtree h = Max( height(k1->getLeft()), k2->getHeight()); k1->setHeight( h+1 ); return k1; } k 1 k 2 Z Y X k 1 k 2 Z Y X singleRightRotation http://ecomputernotes.com
8. TreeNode<int>* singleRightRotation(TreeNode<int>* k2) { if( k2 == NULL ) return NULL; // k1 (first node in k2's left subtree) // will be the new root TreeNode<int>* k1 = k2->getLeft(); // Y moves from k1's right to k2's left k2->setLeft( k1->getRight() ); k1->setRight(k2); // reassign heights. First k2 int h = Max(height(k2->getLeft()), height(k2->getRight())); k2->setHeight( h+1 ); // k2 is now k1's right subtree h = Max( height(k1->getLeft()), k2->getHeight()); k1->setHeight( h+1 ); return k1; } k 1 k 2 Z Y X k 1 k 2 Z Y X singleRightRotation http://ecomputernotes.com
11. TreeNode<int>* singleLeftRotation( TreeNode<int>* k1 ) { if( k1 == NULL ) return NULL; // k2 is now the new root TreeNode<int>* k2 = k1->getRight(); k1->setRight( k2->getLeft() ); // Y k2->setLeft( k1 ); // reassign heights. First k1 (demoted) int h = Max(height(k1->getLeft()), height(k1->getRight())); k1->setHeight( h+1 ); // k1 is now k2's left subtree h = Max( height(k2->getRight()), k1->getHeight()); k2->setHeight( h+1 ); return k2; } k 1 k 2 X Y Z k 1 k 2 X Y Z singleLeftRotation http://ecomputernotes.com
12. TreeNode<int>* singleLeftRotation( TreeNode<int>* k1 ) { if( k1 == NULL ) return NULL; // k2 is now the new root TreeNode<int>* k2 = k1->getRight(); k1->setRight( k2->getLeft() ); // Y k2->setLeft( k1 ); // reassign heights. First k1 (demoted) int h = Max(height(k1->getLeft()), height(k1->getRight())); k1->setHeight( h+1 ); // k1 is now k2's left subtree h = Max( height(k2->getRight()), k1->getHeight()); k2->setHeight( h+1 ); return k2; } k 1 k 2 X Y Z k 1 k 2 X Y Z singleLeftRotation http://ecomputernotes.com
13. TreeNode<int>* singleLeftRotation( TreeNode<int>* k1 ) { if( k1 == NULL ) return NULL; // k2 is now the new root TreeNode<int>* k2 = k1->getRight(); k1->setRight( k2->getLeft() ); // Y k2->setLeft( k1 ); // reassign heights. First k1 (demoted) int h = Max(height(k1->getLeft()), height(k1->getRight())); k1->setHeight( h+1 ); // k1 is now k2's left subtree h = Max( height(k2->getRight()), k1->getHeight()); k2->setHeight( h+1 ); return k2; } k 1 k 2 X Y Z k 1 k 2 X Y Z singleLeftRotation http://ecomputernotes.com
14. TreeNode<int>* singleLeftRotation( TreeNode<int>* k1 ) { if( k1 == NULL ) return NULL; // k2 is now the new root TreeNode<int>* k2 = k1->getRight(); k1->setRight( k2->getLeft() ); // Y k2->setLeft( k1 ); // reassign heights. First k1 (demoted) int h = Max(height(k1->getLeft()), height(k1->getRight())); k1->setHeight( h+1 ); // k1 is now k2's left subtree h = Max( height(k2->getRight()), k1->getHeight()); k2->setHeight( h+1 ); return k2; } k 1 k 2 X Y Z k 1 k 2 X Y Z singleLeftRotation http://ecomputernotes.com
15. TreeNode<int>* singleLeftRotation( TreeNode<int>* k1 ) { if( k1 == NULL ) return NULL; // k2 is now the new root TreeNode<int>* k2 = k1->getRight(); k1->setRight( k2->getLeft() ); // Y k2->setLeft( k1 ); // reassign heights. First k1 (demoted) int h = Max(height(k1->getLeft()), height(k1->getRight())); k1->setHeight( h+1 ); // k1 is now k2's left subtree h = Max( height(k2->getRight()), k1->getHeight()); k2->setHeight( h+1 ); return k2; } k 1 k 2 X Y Z k 1 k 2 X Y Z singleLeftRotation http://ecomputernotes.com
16. TreeNode<int>* singleLeftRotation( TreeNode<int>* k1 ) { if( k1 == NULL ) return NULL; // k2 is now the new root TreeNode<int>* k2 = k1->getRight(); k1->setRight( k2->getLeft() ); // Y k2->setLeft( k1 ); // reassign heights. First k1 (demoted) int h = Max(height(k1->getLeft()), height(k1->getRight())); k1->setHeight( h+1 ); // k1 is now k2's left subtree h = Max( height(k2->getRight()), k1->getHeight()); k2->setHeight( h+1 ); return k2; } k 1 k 2 X Y Z k 1 k 2 X Y Z singleLeftRotation http://ecomputernotes.com
17. TreeNode<int>* doubleRightLeftRotation(TreeNode<int>* k1) { if( k1 == NULL ) return NULL; // single right rotate with k3 (k1's right child) k1->setRight( singleRightRotation(k1->getRight())); // now single left rotate with k1 as the root return singleLeftRotation(k1); } k 1 k 3 D A B C k 2 k 1 k 3 D A B C k 2 doubleRightLeftRotation http://ecomputernotes.com
18. TreeNode<int>* doubleRightLeftRotation(TreeNode<int>* k1) { if( k1 == NULL ) return NULL; // single right rotate with k3 (k1's right child) k1->setRight( singleRightRotation(k1->getRight())); // now single left rotate with k1 as the root return singleLeftRotation(k1); } k 1 k 3 D A B C k 2 k 1 k 3 D A B C k 2 doubleRightLeftRotation http://ecomputernotes.com
19. TreeNode<int>* doubleRightLeftRotation(TreeNode<int>* k1) { if( k1 == NULL ) return NULL; // single right rotate with k3 (k1's right child) k1->setRight( singleRightRotation(k1->getRight())); // now single left rotate with k1 as the root return singleLeftRotation(k1); } k 1 k 2 D A B C k 3 k 1 k 3 D A B C k 2 doubleRightLeftRotation http://ecomputernotes.com
20. TreeNode<int>* doubleLeftRightRotation(TreeNode<int>* k3) { if( k3 == NULL ) return NULL; // single left rotate with k1 (k3's left child) k3->setLeft( singleLeftRotation(k3->getLeft())); // now single right rotate with k3 as the root return singleRightRotation(k3); } k 1 k 3 D A B C k 2 k 1 k 3 D A B C k 2 doubleRightLeftRotation http://ecomputernotes.com
21. TreeNode<int>* doubleLeftRightRotation(TreeNode<int>* k3) { if( k3 == NULL ) return NULL; // single left rotate with k1 (k3's left child) k3->setLeft( singleLeftRotation(k3->getLeft())); // now single right rotate with k3 as the root return singleRightRotation(k3); } k 1 k 3 D A B C k 2 k 1 k 3 D A B C k 2 doubleRightLeftRotation http://ecomputernotes.com
22. TreeNode<int>* doubleLeftRightRotation(TreeNode<int>* k3) { if( k3 == NULL ) return NULL; // single left rotate with k1 (k3's left child) k3->setLeft( singleLeftRotation(k3->getLeft())); // now single right rotate with k3 as the root return singleRightRotation(k3); } k 1 k 3 D A B C k 2 k 1 k 3 D A B C k 2 doubleRightLeftRotation http://ecomputernotes.com
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33. Deletion in AVL Tree Case 1a : the parent of the deleted node had a balance of 0 and the node was deleted in the parent’s left subtree. Action : change the balance of the parent node and stop. No further effect on balance of any higher node. Delete on this side http://ecomputernotes.com
34. Deletion in AVL Tree Here is why; the height of left tree does not change. 1 2 3 4 5 6 7 0 1 2
35. Deletion in AVL Tree Here is why; the height of left tree does not change. 1 2 3 4 5 6 7 2 3 4 5 6 7 0 1 2 remove(1)
36. Deletion in AVL Tree Case 1b : the parent of the deleted node had a balance of 0 and the node was deleted in the parent’s right subtree. Action : (same as 1a ) change the balance of the parent node and stop. No further effect on balance of any higher node. Delete on this side
37. Deletion in AVL Tree Case 2a : the parent of the deleted node had a balance of 1 and the node was deleted in the parent’s left subtree. Action : change the balance of the parent node. May have caused imbalance in higher nodes so continue up the tree. Delete on this side
