Self-Balancing Trees: AVL and Red-Black Trees

Learning Objectives Explain why self-balancing is critical for maintaining the efficiency of Binary Search Trees (BSTs). Define and calculate the 'balance factor' of any node within an AVL tree. Trace the insertion process in an AVL tree, correctly identifying and performing the four rotation types (LL, RR, LR, RL) to restore balance. List and describe the five core properties that define a valid Red-Black Tree. Compare and contrast the performance characteristics and typical use cases of AVL trees versus Red-Black trees. Analyze and state the time complexity for search, insert, and delete operations in self-balancing trees, explaining why it is O(log n). Ever wondered how a database can find one specific record out of a billion in a fraction of a second? 🤔 It&#03...

TermDefinitionExample Balance Factor (AVL Trees)The difference between the height of the right subtree and the height of the left subtree of a node. It is calculated as: height(right subtree) - height(left subtree).If a node's left subtree has a height of 2 and its right subtree has a height of 1, its balance factor is 1 - 2 = -1. AVL TreeA self-balancing Binary Search Tree where the balance factor of every node must be -1, 0, or 1. If an insertion or deletion causes a node's balance factor to become -2 or +2, the tree performs rotations to restore balance.A tree with root 10, left child 5, and right child 15 is a valid AVL tree because all nodes have a balance factor of 0. Tree RotationA fundamental operation used to change the structure of a tree while preserving the Binary Se...

AVL Tree Balance Property | height(node.right) - height(node.left) | ≤ 1 This is the defining rule of an AVL tree. For every single node in the tree, the height of its two child subtrees can differ by at most one. This property must be checked and restored via rotations after every insertion or deletion. Red-Black Tree Properties 1. Every node is Red or Black. 2. The root is always Black. 3. A Red node cannot have a Red child (no two adjacent Red nodes). 4. Every path from a node to any of its descendant NULL leaves contains the same number of Black nodes (the 'black-height'). These five rules work together to ensure that the longest path from the root to a leaf is no more than twice as long as the shortest path. This provides the guarantee of O(log n) performance....