Process

Definition

Sequence of interdependent and linked procedures which, at every stage, consume one or more resources (employee time, energy, machines, money) to convert inputs (data, material, parts, etc.) into outputs. These outputs then serve as inputs for the next stage until a known goal or end result is reached.

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 (http://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.parlante@cs.stanford.edu.

Related CSLibrary Articles

• Linked List Problems (http://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 (http://cslibrary.stanford.edu/102/) -- basic concepts of pointers and memory
• The Great Tree-List Problem (http://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.

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 (<=), and all the elements in its right subtree are greater than 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, and its right subtree nodes (6, 9) 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 and 4 > 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 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 (http://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.parlante@cs.stanford.edu.

Related CSLibrary Articles

• Linked List Problems (http://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 (http://cslibrary.stanford.edu/102/) -- basic concepts of pointers and memory
• The Great Tree-List Problem (http://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.

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 (<=), and all the elements in its right subtree are greater than 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, and its right subtree nodes (6, 9) 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 and 4 > 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

In computer science, a binary search tree (BST) is a node based binary tree data structure which has the following properties:^[1]

• The left subtree of a node contains only nodes with keys less than the node's key.
• The right subtree of a node contains only nodes with keys greater than the node's key.
• Both the left and right subtrees must also be binary search trees.

From the above properties it naturally follows that:

• Each node (item in the tree) has a distinct key.

Generally, the information represented by each node is a record rather than a single data element. However, for sequencing purposes, nodes are compared according to their keys rather than any part of their associated records.

The major advantage of binary search trees over other data structures is that the related sorting algorithms and search algorithms such as in-order traversal can be very efficient.

Binary search trees are a fundamental data structure used to construct more abstract data structures such as sets, multisets, and associative arrays.