# Binary Tree

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Binary Trees Page: 1 Binary Trees by Nick Parlante This article introduces the basic concepts of binary trees, and then works through a series of practice problems with solution code in C/C++ and Java. Binary trees have an elegant recursive pointer structure, so they are a good way to learn recursive pointer algorithms. Contents Section 1. Binary Tree Structure — a quick introduction to binary trees and the code that operates on them Section 2. Binary Tree Problems — practice problems in increasing order of difficulty Section 3. C Solutions — solution code to the problems for C and C++ programmers Section 4. Java versions — how binary trees work in Java, with solution code Stanford CS Education Library — #110 This is article #110 in the Stanford CS Education Library. This and other free CS materials are available at the library (https://cslibrary. stanford. edu/). That people seeking education should have the opportunity to find it. This article may be used, reproduced, excerpted, or sold so long as this paragraph is clearly reproduced. Copyright 2000-2001, Nick Parlante, nick. [email protected] stanford. edu. Related CSLibrary Articles Linked List Problems (https://cslibrary. stanford. edu/105/) — a large collection of linked list problems using various pointer techniques (while this binary tree article concentrates on recursion) Pointer and Memory (https://cslibrary. stanford. edu/102/) — basic concepts of pointers and memory The Great Tree-List Problem (https://cslibrary. stanford. edu/109/) — a great pointer recursion problem that uses both trees and lists Section 1 — Introduction To Binary Trees A binary tree is made of nodes, where each node contains a “left” pointer, a “right” pointer, and a data element. The “root” pointer points to the topmost node in the tree. The left and right pointers recursively point to smaller “subtrees” on either side. A null pointer represents a binary tree with no elements — the empty tree. The formal recursive definition is: a binary tree is either empty (represented by a null pointer), or is made of a single node, where the left and right pointers (recursive definition ahead) each point to a binary tree. ttp://cslibrary. stanford. edu/110/ BinaryTrees. html Binary Trees Page: 2 A “binary search tree” (BST) or “ordered binary tree” is a type of binary tree where the nodes are arranged in order: for each node, all elements in its left subtree are less-or-equal to the node (). The tree shown above is a binary search tree — the “root” node is a 5, and its left subtree nodes (1, 3, 4) are 5. Recursively, each of the subtrees must also obey the binary search tree constraint: in the (1, 3, 4) subtree, the 3 is the root, the 1 3. Watch out for the exact wording in the problems — a “binary search tree” is different from a “binary tree”. The nodes at the bottom edge of the tree have empty subtrees and are called “leaf” nodes (1, 4, 6) while the others are “internal” nodes (3, 5, 9). Binary Search Tree Niche Basically, binary search trees are fast at insert and lookup.