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Balanced Binary Search Trees

Learn about balanced binary search trees, including AVL trees and red-black trees, and how to insert and search nodes efficiently.

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Balanced Binary Search Trees

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  1. Balanced Binary Search Trees • Pathological BST • Insert nodes from ordered list : O(___) ? • Search: O(___) ? • The Balanced Tree • Binary Tree is balanced if height of left and right subtree differ by no more than one, recursively for all nodes. • (Height of empty tree is -1) • Examples

  2. Balanced Binary Search Trees • Keeping BSTrees Balanced • Keeps find, insert, delete O(log(N)) worst case. • Pay small extra amount at each insertion to keep it balanced • Several Well-known Systems Exist for This • AVL Trees • Red-Black Trees • . . . • Will “look at” AVL Trees

  3. AVL Trees • AVL Trees • Adelson-Velskii and Landis • Discovered ways to keep BSTrees Balanced • Insertions • Insert into BST in normal way • If tree no longer balanced, perform “rotation(s)” • Rotations restore balance to the tree

  4. AVL Trees • Single Rotation • An insertion into the left subtree of the left child of tree • Adapted from Weiss, pp 567-568 /** Used if insert has caused loss of balance at k2 * (Also used as part of double rotation operations) * @return root of adjusted tree */ TNode rotateWithLeftChild(TNode k2){ TNode k1 = k2.left; k2.left = k1.right; k1.right = k2; return k1; }

  5. AVL Trees • Single Rotation

  6. AVL Trees • Single Rotation k1 k2 before after k2 k1 C B B C A A • Also: mirror image

  7. AVL Trees • Single Rotation • Mirror image case /** Used if insert has caused loss of balance at k2 * (Also used as part of double rotation operations) * @return root of adjusted tree */ TNode rotateWithRightChild(TNode k2){ TNode k1 = k2.right; k2.right = k1.left; k1.left = k2; return k1; }

  8. AVL Tree • Double Rotation • An insertion into the right subtree of the left child of tree • Adapted from Weiss, p 57 /** Used after insertion into right subtree, k2, * of left child, k1, of k3 (if it has caused * loss of balance) * @return root of adjusted tree */ TNode doubleRotateWithLeftChild(TNode k3) { k3.left = rotateWithRightChild(k3.left); return rotateWithLeftChild(k3); }

  9. AVL Tree • Double Rotation

  10. AVL Trees • Double Rotation k3 after before k2 k3 k1 k1 k2 C B D A D A B C • Also: mirror image

  11. AVL Tree • Double Rotation • An insertion into the right subtree of the left child of tree • Adapted from Weiss, p 571 /** Used after insertion into right subtree, k2, * of right child, k1, of k3 (if it has caused * loss of balance) * @return root of adjusted tree */ TNode doubleRotateWithRightChild(TNode k3) { k3.right = rotateWithLeftChild(k3.right); return rotateWithRightChild(k3); }

  12. AVL Trees • Deletions can also cause imbalance • Use similar rotations to restore balance • Big Oh? • That was the “big picture” • How can we insure performance? • Think about some of the implementation issues.

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