Balanced Binary Search Trees

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Presentation transcript:

Balanced Binary Search Trees CS-2851 Dr. Mark L. Hornick

For any balanced binary search tree, searching, inserting and deleting have worstTime(n) that is logarithmic in n But insertions and deletions can result in a tree that is highly unbalanced O(log2 n) performance drifts toward O(n) A BST is balanced only if if its height is logarithmic in n such that H = 1+Math.floor(log2(n)) N=8, h= 1+floor(log2(8)) N=5, h > 1+floor(log2(5) ) CS-2851 Dr. Mark L. Hornick

An AVL tree is a balanced BST that either is empty or in which… Heights between left and right subtrees differs by only 1 The left and right subtrees are also AVL trees. CS-2851 Dr. Mark L. Hornick

Another (balanced) AVL Tree 20 30 10 35 5 32 CS-2851 Dr. Mark L. Hornick

More AVL Tree examples CS-2851 Dr. Mark L. Hornick

Non-AVL Tree examples CS-2851 Dr. Mark L. Hornick

Balance can be maintained through rotations Rotation: an adjustment to the tree, around an element, that maintains the required ordering of elements CS-2851 Dr. Mark L. Hornick

For any right rotation around element x, the right subtree of x’s left child becomes the left subtree of x. CS-2851 Dr. Mark L. Hornick

CS-2851 Dr. Mark L. Hornick

For any right rotation around element x, the right subtree of x’s left child becomes the left subtree of x. CS-2851 Dr. Mark L. Hornick

Exercise: Rotate right about 20 Rule: for any right rotation around element x, the right subtree of x’s left child becomes the left subtree of x. 20 35 10 5 12 7 CS-2851 Dr. Mark L. Hornick

Solution: Right rotation about 20 10 35 10 20 5 5 12 35 12 7 7 Note improvement in height CS-2851 Dr. Mark L. Hornick

Exercise: Rotate right about 10 Rule: for any right rotation around element x, the right subtree of x’s left child becomes the left subtree of x. 20 35 10 5 12 7 CS-2851 Dr. Mark L. Hornick

Solution: Right rotation about 10 20 20 5 35 10 35 10 5 12 12 7 7 Note NO improvement in height CS-2851 Dr. Mark L. Hornick

Exercise: Rotate left about 10 Rule: for any left rotation around element x, the left subtree of x’s right child becomes the right subtree of x. 20 35 10 12 5 11 13 7 CS-2851 Dr. Mark L. Hornick

Solution: Left rotation about 10 20 20 35 12 10 35 10 13 5 12 5 11 11 13 7 7 Note increase in height CS-2851 Dr. Mark L. Hornick

In a rotation around x, the only restructuring is to the subtree rooted at x (nothing above x is affected) CS-2851 Dr. Mark L. Hornick

Application of rotations in a certain way maintains the balance of Binary Search Trees Resulting in AVL trees Red/Black trees others… CS-2851 Dr. Mark L. Hornick

Rotations summary CS-2851 Dr. Mark L. Hornick