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Understanding Balanced Trees: AVL and Red-Black Trees

Learn about AVL trees, including balancing techniques, rotations, insertion, deletion, and efficiency. Explore Red-Black trees and other balanced tree varieties like α-balanced trees.

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Understanding Balanced Trees: AVL and Red-Black Trees

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  1. Balanced Trees • There are several ways to define balance • Examples: • Force the subtrees of each node to have almost equal heights • Place upper and lower bounds on the heights of the subtrees of each node. • Force the subtrees of each node to have similar sizes (=number of nodes)

  2. AVL Trees • AVL tree = • A binary search tree • with the property: • for every node, the heights of the left and right subtrees differ at most by one. • Implementation issues: • Each node contains a value (-1, 1, 0) indicating which subtree is "heavier" • Insert and Delete are modified. They restructure the tree to make it balanced (if necessary).

  3. AVL trees 1 nodes are marked with balance value -1 1 0 1 0 0

  4. -2 2 -2 2 1 -1 -1 1 0 0 0 0 AVL trees: Fixing imbalances • An imbalance is detected when the height difference between two subtrees of a node becomes greater than 1 or smaller than -1. • There are two types of imbalances: or or TYPE 1 TYPE 2

  5. AVL trees: Fixing imbalances • Fixing an imbalance is done by rotating the tree. • There are two types of rotation: • single rotation • for TYPE 1 imbalances • double rotation • for TYPE 2 imbalances • consists of two single rotations. • The rotations must always preserve the BST property.

  6. AVL Trees: single rotation node with imbalance 6 D 4 4 right rotate at node 6 C 2 6 2 A B C D B A This is a single right rotation. A single left rotation is symmetric.

  7. AVL Trees: Double rotation node with imbalance node with imbalance 6 6 4 D D 4 2 2 6 STEP2: right rotate at node 6 C A A B C D 2 4 B A C B STEP 1: left rotate at node 2

  8. AVL Trees: Double rotation If you want to do it in one step, imagine taking 4 and moving it up in between 2 and 6, so that 2 and 6 become its new children: node with imbalance 6 D 4 2 2 6 A 4 A B C D C B B and C will then be adopted by 2 and 6 respectively, in order to maintain the BST property.

  9. AVL Trees: Insert • Insert the node as in a BST • Starting at the newly inserted node, travel up the tree (towards the root), updating the balances along the way. • If the tree becomes unbalanced, decide which rotation needs to be performed, rotate the tree and update the balances.

  10. AVL Trees: Delete • Delete the node as in a BST • Starting at the newly inserted node, travel up the tree (towards the root), updating the balances along the way. • If the tree becomes unbalanced, decide which rotation needs to be performed, rotate the tree and update the balances. • Keep traveling towards the root, checking the balances. You may need to rotate again.

  11. AVL Trees: Insert/Delete • Insert • maximum possible number of rotations = 1 • Delete • maximum possible number of rotations = lgn • Worst case times • search: O(lgn) • insert: O(lgn) • delete: O(lgn) • one rotation: O(1)

  12. AVL Trees: Efficiency • It can be shown that the worst case height of an AVL tree is at most 44% larger than the minimum possible for a BST (i.e. approximately 1.44lgn)

  13. Other balanced trees • Red-black trees • Each node has an extra bit to hold a color • The tree stays balanced by placing restrictions on the way the nodes are colored • Every path from the root to a leaf has the same number of black nodes and there cannot be consecutive red nodes. • A red-black tree is balanced when the longest path from the root to a leaf is at most twice the length of the shortest path from the root to a leaf. • A red-black tree with n nodes has height at most 2lg(n+1)

  14. Other balanced trees • α-balanced trees • α is a constant between 1/2 and 1 • A tree is α-balanced if, for every node x:size[left(x)]  α·size[x]size[right(x)]  α·size[x]

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