A balanced tree is a tree data structure in which the height of the left and right subtrees of any node differs by at most one. This ensures that the tree remains approximately balanced, preventing one branch from becoming significantly deeper than another. The goal of maintaining balance is to optimize the performance of certain operations on the tree, such as searching, insertion, and deletion.
Balanced trees are particularly important in scenarios where the efficiency of these operations is crucial, as an unbalanced tree can lead to degenerate cases where the time complexity of operations becomes inefficient, approaching O(n) instead of the more desirable O(log n).
One common type of balanced tree is the AVL tree (Adelson-Velsky and Landis tree), which is a self-balancing binary search tree. In an AVL tree, each node is assigned a balance factor, which is the height of the right subtree minus the height of the left subtree. The balance factor is required to be in the range [-1, 0, 1] for every node. If, at any point during insertion or deletion, the balance factor of a node violates this range, rotations are performed to restore balance.
Balanced trees, including AVL trees and Red-Black trees, are commonly used in applications where maintaining a predictable and efficient time complexity for various operations is essential. Examples include database indexing, compiler symbol tables, and other scenarios where efficient search, insertion, and deletion operations are crucial. These trees ensure that the height of the tree remains logarithmic with respect to the number of elements, leading to better overall performance.
