Binary trees and Binary Search Trees (BSTs) form the foundation of hierarchical data structures. A BST maintains the invariant that left children are less than the parent and right children are greater, enabling O(log n) average-case search and insert. Tree traversals come in three primary flavors: inorder (Left-Root-Right) yields sorted order for BSTs, preorder (Root-Left-Right) is used for serialization, and postorder (Left-Right-Root) is ideal for deletion operations. Level-order traversal uses BFS to visit nodes level by level, with variants like zigzag traversal alternating direction.
Advanced tree operations include finding the Lowest Common Ancestor (LCA), validating BST property, computing tree diameter, and detecting balanced height. The LCA problem has an elegant recursive solution for general binary trees, while BSTs leverage their sorted property for an even simpler iterative approach. The Maximum Path Sum problem requires careful tracking because the optimal path may pass through or end at any node. Tree construction from traversals combines preorder and inorder arrays with hash maps for O(n) reconstruction.
Specialized tree structures extend these concepts in powerful ways. The Trie (Prefix Tree) stores strings character by character, enabling O(m) prefix operations used in autocomplete and spell check. The Segment Tree and Binary Indexed Tree (Fenwick Tree) support efficient range queries with O(log n) operations, with the segment tree's lazy propagation variant extending this to range updates. More exotic structures like AVL trees, red-black trees, and treaps maintain balance to guarantee O(log n) operations. Morris Traversal achieves O(1) space tree traversal by using threaded binary trees, temporarily linking rightmost nodes back to ancestors.