Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
expand_more
expand_more
format_list_bulleted
Question
Chapter 22.1, Problem 2E
Program Plan Intro
To give an adjacency-list representation for a complete binary tree on 7 vertices and an equivalent adjacency-matrix representation.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
A complete binary tree is a graph defined through the following recursive definition. Basis step: A single vertex is a complete binary tree. Inductive step: If T1 and T2 are disjoint complete binary trees with roots r1, r2, respectively, the the graph formed by starting with a root r, and adding an edge from r to each of the vertices r1, r2 is also a complete binary tree. The set of leaves of a complete binary tree can also be defined recursively. Basis step: The root r is a leaf of the complete binary tree with exactly one vertex r. Inductive step: The set of leaves of the tree T built from trees T1, T2 is the union of the sets of leaves of T1 and the set of leaves of T2.
The height h(T) of a binary tree is defined in the class. Use structural induction to show that `(T), the number of leaves of a complete binary tree T, satisfies the following inequality `(T) ≤ 2 h(T) .
Draw the binary search tree if the in-order traversal of that binary search tree are given as below:
In-order traversal: 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 19, 20
6*. Show that there is a unique binary tree with five vertices A, B, C, D, E that has preorder
traversal ABCDE and inorder traversal ADECB.
Chapter 22 Solutions
Introduction to Algorithms
Ch. 22.1 - Prob. 1ECh. 22.1 - Prob. 2ECh. 22.1 - Prob. 3ECh. 22.1 - Prob. 4ECh. 22.1 - Prob. 5ECh. 22.1 - Prob. 6ECh. 22.1 - Prob. 7ECh. 22.1 - Prob. 8ECh. 22.2 - Prob. 1ECh. 22.2 - Prob. 2E
Ch. 22.2 - Prob. 3ECh. 22.2 - Prob. 4ECh. 22.2 - Prob. 5ECh. 22.2 - Prob. 6ECh. 22.2 - Prob. 7ECh. 22.2 - Prob. 8ECh. 22.2 - Prob. 9ECh. 22.3 - Prob. 1ECh. 22.3 - Prob. 2ECh. 22.3 - Prob. 3ECh. 22.3 - Prob. 4ECh. 22.3 - Prob. 5ECh. 22.3 - Prob. 6ECh. 22.3 - Prob. 7ECh. 22.3 - Prob. 8ECh. 22.3 - Prob. 9ECh. 22.3 - Prob. 10ECh. 22.3 - Prob. 11ECh. 22.3 - Prob. 12ECh. 22.3 - Prob. 13ECh. 22.4 - Prob. 1ECh. 22.4 - Prob. 2ECh. 22.4 - Prob. 3ECh. 22.4 - Prob. 4ECh. 22.4 - Prob. 5ECh. 22.5 - Prob. 1ECh. 22.5 - Prob. 2ECh. 22.5 - Prob. 3ECh. 22.5 - Prob. 4ECh. 22.5 - Prob. 5ECh. 22.5 - Prob. 6ECh. 22.5 - Prob. 7ECh. 22 - Prob. 1PCh. 22 - Prob. 2PCh. 22 - Prob. 3PCh. 22 - Prob. 4P
Knowledge Booster
Similar questions
- Kruskal's algorithm, which incorporates the union-find data structure, is as follows.algorithm Pre-cond KruskalMST (G): G is an undirected graph.post-cond: The result is a minimum spanning tree.arrow_forward2. Use a deep-first search to find a spanning tree of the following graph starting from vertex а. (a) Write the list of the edges of the spanning tree in the order you add them. (b) Draw the minimal spanning tree.arrow_forwardA complete binary tree is a graph defined through the following recur- sive definition. Basis step: A single vertex is a complete binary tree.Inductive step: If T1 and T2 are disjoint complete binary trees with roots r1, r2, respectively, the graph formed by starting with a root r, and adding an edge from r to each of the vertices r1,r2 is also a complete binary tree.The set of leaves of a complete binary tree can also be defined recursively.Basis step: The root r is a leaf of the complete binary tree with exactly one vertex r.Inductive step: The set of leaves of the tree T built from trees T1, T2 is the union of the sets of leaves of T1 and the set of leaves of T2. The height h(T ) of a binary tree is defined in the class.Use structural induction to show that L(T), the number of leaves of a complete binary tree T , satisfies the following inequality L(T) ≤ 2^h(T).arrow_forward
- Use Prim’s algorithm starting at node A to compute theMinimum Spanning Tree (MST) of the following graph. Inparticular, write down the edges of the MST in the order inwhich Prim’s algorithm adds them to the MST. Use theformat (node1; node2) to denote an edge.arrow_forwardAdjacency-lists data structure. The standard graph representation for graphs that arenot dense is called the adjacency-lists data structure, where we keep track of all thevertices adjacent to each vertex on a linked list that is associated with that vertex. Wemaintain an array of lists so that, given a vertex, we can immediately access its list. Toimplement lists, we use our Bag ADT from Section 1.3 with a linked-list implementation, so that we can add new edges in constant time and iterate through adjacent vertices in constant time per adjacent vertex. The Graph implementation on page 526 is basedon this approach, and the figure on the facing page depicts the data structures built bythis code for tinyG.txt. To add an edge connecting v and w, we add w to v’s adjacencylist and v to w’s adjacency list. Thus, each edge appears twice in the data structure. ThisGraph implementation achieves the following performance characteristics:■ Space usage proportional to V + E■ Constant time to add…arrow_forward1. A complete binary tree of depth d is a tree where every node has two children, except for the nodes at depth d, which have no children. Such a tree has 2' nodes at depth j, where j = 0, 1, d. What is the minimal size of a vertex cover of a complete binary tree of depth 2m + 1, where m is a nonnegative integer?arrow_forward
- A complete binary tree is a graph defined through the following recur- sive definition. Basis step: A single vertex is a complete binary tree.Inductive step: If T1 and T2 are disjoint complete binary trees with roots r1, r2, respectively, the the graph formed by starting with a root r, and adding an edge from r to each of the vertices r1,r2 is also a complete binary tree.The set of leaves of a complete binary tree can also be defined recur- sively.Basis step: The root r is a leaf of the complete binary tree with exactly one vertex r.Inductive step: The set of leaves of the tree T built from trees T1, T2 is the union of the sets of leaves of T1 and the set of leaves of T2. The height h(T ) of a binary tree is defined in the class.Use structural induction to show that L(T), the number of leaves of a complete binary tree T , satisfies the following inequality L(T) ≤ 2^h(T).arrow_forwardGive an adjacency matrix and an adjacency list for the following graphs: 1. A complete binary tree with 7 nodes arranged in level order (like the indices of a heaps) are numbered consecutively. 2. A complete graph with 4 nodes.arrow_forwardLet B be a binary tree. If for each of its vertices v the data item inserted in v is greater than the data item inserted in the left son of the vertex v and at the same time smaller than the data item inserted in the right son of the vertex v, then B is a search tree. Prove that if G is a tree, then its vertex with maximum eccentricity is a leaf.arrow_forward
- Kruskal’s algorithm with the union–find data structure incorporated is as follows.algorithm KruskalMST (G) pre-cond: G is an undirected graph. post-cond: The output consists of a minimal spanning tree.arrow_forwardConstruct two different ordered trees of 8 vertices whose postorder traversal produces the same list. With explanation please.arrow_forwardWrite Kruskal’s algorithm with the union–find data structure incorporated is as follows. pre-cond: G is an undirected graph. post-cond: The output consists of a minimal spanning tree.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Database System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSON
- C How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSONDatabase Systems: Design, Implementation, & Manag...Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage LearningProgrammable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill Education
Database System Concepts
Computer Science
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:9780134444321
Author:Tony Gaddis
Publisher:PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:9780132737968
Author:Thomas L. Floyd
Publisher:PEARSON
C How to Program (8th Edition)
Computer Science
ISBN:9780133976892
Author:Paul J. Deitel, Harvey Deitel
Publisher:PEARSON
Database Systems: Design, Implementation, & Manag...
Computer Science
ISBN:9781337627900
Author:Carlos Coronel, Steven Morris
Publisher:Cengage Learning
Programmable Logic Controllers
Computer Science
ISBN:9780073373843
Author:Frank D. Petruzella
Publisher:McGraw-Hill Education