Tutorial Playlist

Data Structure Tutorial

Overview

Arrays in Data Structures: A Guide With Examples

Lesson - 1

All You Need to Know About Two-Dimensional Arrays

Lesson - 2

All You Need to Know About a Linked List in a Data Structure

Lesson - 3

The Complete Guide to Implement a Singly Linked List

Lesson - 4

The Ultimate Guide to Implement a Doubly Linked List

Lesson - 5

The Fundamentals for Understanding Circular Linked List

Lesson - 6

The Ultimate Guide To Understand The Differences Between Stack And Queue

Lesson - 7

Implementing Stacks in Data Structures

Lesson - 8

Your One-Stop Solution for Stack Implementation Using Array

Lesson - 9

Your One-Stop Solution for Queue Implementation Using Array

Lesson - 10

Your One-Stop Solution to Learn Depth-First Search(DFS) Algorithm From Scratch

Lesson - 11

Your One-Stop Solution for Stack Implementation Using Linked-List

Lesson - 12

The Definitive Guide to Understand Stack vs Heap Memory Allocation

Lesson - 13

All You Need to Know About Linear Search Algorithm

Lesson - 14

All You Need to Know About Breadth-First Search Algorithm

Lesson - 15

A One-Stop Solution for Using Binary Search Trees in Data Structure

Lesson - 16

The Best Tutorial to Understand Trees in Data Structure

Lesson - 17

A Complete Guide to Implement Binary Tree in Data Structure

Lesson - 18

A Holistic Look at Using AVL Trees in Data Structures

Lesson - 19

All You Need to Know About Tree Traversal in Data Structure

Lesson - 20

The Best Guide You’ll Ever Need to Understand B-Tree in Data Structure

Lesson - 21

The Best Guide You'll Ever Need to Understand Spanning Tree in Data Structure

Lesson - 22

The Best and Easiest Way to Understand an Algorithm

Lesson - 23

Your One-Stop Solution to Understand Shell Sort Algorithm

Lesson - 24

Your One-Stop Solution to Quick Sort Algorithm

Lesson - 25

The Most Useful Guide to Learn Selection Sort Algorithm

Lesson - 26

Everything You Need to Know About Radix Sort Algorithm

Lesson - 27

Everything You Need to Know About the Counting Sort Algorithm

Lesson - 28

Everything You Need to Know About the Merge Sort Algorithm

Lesson - 29

Insertion Sort Algorithm: One-Stop Solution That Will Help You Understand Insertion Sort

Lesson - 30

Everything You Need to Know About the Bubble Sort Algorithm

Lesson - 31

The Best Guide You’ll Ever Need to Understand Bucket Sort Algorithm

Lesson - 32

Your One-Stop Solution to Understand Recursive Algorithm in Programming

Lesson - 33

The Definitive Guide to Understanding Greedy Algorithm

Lesson - 34

Your One-Stop Solution to Understand Backtracking Algorithm

Lesson - 35

The Fundamentals of the Bellman-Ford Algorithm

Lesson - 36

Your One-Stop Solution for Graphs in Data Structures

Lesson - 37

The Best Guide to Understand and Implement Solutions for Tower of Hanoi Puzzle

Lesson - 38

A Simplified and Complete Guide to Learn Space and Time Complexity

Lesson - 39

All You Need to Know About the Knapsack Problem : Your Complete Guide

Lesson - 40

The Fibonacci Series: Mathematical and Programming Interpretation

Lesson - 41

The Holistic Look at Longest Common Subsequence Problem

Lesson - 42

The Best Article to Understand What Is Dynamic Programming

Lesson - 43

A Guide to Implement Longest Increasing Subsequence Using Dynamic Programming

Lesson - 44

A Holistic Guide to Learn Stop Solution Using Dynamic Programming

Lesson - 45

One Stop Solution to All the Dynamic Programming Problems

Lesson - 46

Understanding the Fundamentals of Binomial Distribution

Lesson - 47
The Best Guide You'll Ever Need to Understand Spanning Tree in Data Structure

Did you know that in a Local Area Network, the spanning tree data structure is applied to manage routing systems? Additionally, it is also used to implement telecommunication networks, transportation networks, and electrical grids as it provides an optimal implementation path. So, in this article, we will discover what a spanning tree in data structure is and understand its functionalities and applications.

Graphs and Their Different Types

Spanning tree in data structures and algorithms is developed by referencing the mathematical field of graph theory. Thus, primarily, we shall understand a few terminologies about the graph at a glance. 

A graph is a structure that contains vertices and edges connecting them. Based on their edge connectivity, there are three basic types of graphs as follows:

  • Undirected Graph

The graph in which all the edges don’t point to any specific direction is called an undirected graph. Since there is no particular direction given for traversal, this graph’s edges are considered bidirectional. The representation of the graph shown below is an example of an undirected graph.

Undirected_Graph_Illustration

  • Directed Graph

The graph in which all the edges point to only one specific direction is called a directed graph. In this type of graph, the retrieval to the last node is not possible since the edges of this graph can only be traversed in one direction.

Directed_graph_illustration.

In the diagram shown above, you can clearly observe that the traversal is possible from direction A -> E.However, you cannot retrieve back at position A from node E due to the directions of edges.

Full Stack Web Developer Course

To become an expert in MEAN StackView Course
Full Stack Web Developer Course

  • Connected Graph

A connected graph is a graph in which there is a path from one vertex to any other vertex in a graph. According to this definition, null graphs and singleton graphs can also be called connected graphs. The graph shown below is a connected graph, as you can visit any vertex from any other vertex of a graph.

Connected_Graph_Illustration.

Introduction to Spanning Tree

If you have graph G with vertices V and edges E, then that graph can be represented as G(V, E). For this graph G(V, E), if you construct a tree structure G’(V’, E’) such that the formed tree structure follows constraints mentioned below, then that structure can be called a Spanning Tree. 

  1. V’ = V   (number of Vertices in G’ must be equal to the number of vertices in G)
  2. E’ = |V| - 1   (Edges of G’ must be equal to the number of vertices in graph G minus 1)

Let’s understand this by creating spanning trees for a particular graph structure. 

Creating Spanning Trees for Given Graph

Let’s assume that you want to create the spanning tree structures for the graph given below:

Graph_G.

As mentioned earlier, the spanning tree has the same number of vertices as the graph. In this case, you have a total number of vertices in the graph equal to 5. Thus, T(V’, E’) will also have 5 vertices in its structure. Additionally,  the number of edges E’ must be equal to the number of vertices in the graph minus one, i.e., 4. For this given graph, five spanning trees can be constructed as shown below:

Spanning_tree_structures_for_Graph-G

How to Calculate the Number of Possible Spanning Trees

A connected graph can have several spanning trees, as previously stated. So, how do you estimate how many distinct spanning trees can be created for a given graph? Graph theory in mathematics provides the answer to this question. According to graph theory, in order to determine the number of feasible spanning trees, you must first determine the graph's type.

If a given graph formulates a closed cycle and has the number of vertices equal to the number of edges, then that graph can be called a cycle graph. And the number of possible spanning trees for any cycle graph is equal to the number of its vertices or edges. The graph for which we created the spanning trees previously had 5 vertices and 5 edges with a closed cycle. 

Thus, n(ST)cycle graph =V =E =5

Otherwise, if a unique edge connects each pair of vertices in a graph, it will be considered as a complete graph. And the number of possible spanning trees for this complete graph can be calculated using Cayley’s Formula:

 n(ST)complete graph =V(v-2)

The graph given below is an example of a complete graph consisting of 4 vertices and 6 edges. For this graph, number of possible spanning trees will be:

Complete_Graph

 n(ST)cg =V(v-2)=4(4-2)=42=16

Properties of Spanning Tree

In parallel and distributed computing, spanning trees are crucial. Listed below are a few important properties of spanning trees.

  • A spanning tree whose overall resultant weight value is minimal is considered to be a Minimal Spanning Tree.
  • A connected graph can have more than one spanning tree.
  • All Spanning trees must contain the same number of vertices as of graph, and the number of edges must be equal to |V| - 1.
  • The spanning tree must not contain any cycle.
  • If the given graph is a cycle graph, then the number of possible spanning trees will be equal to the number of vertices of the given graph.
  • If the given graph is a complete graph, then the number of possible spanning trees can be calculated using Cayley’s Formula.
  • A spanning tree cannot be disconnected. If you remove any edge from the created tree, then it won’t be considered as a spanning tree anymore. It can only be regarded as a disconnected graph. The diagram given below explains the same:

Spanning_Tree_Property.

  • There is a chance that there will be more than one minimum spanning tree if there are numerous edges with the same weight.

Consider the example given below. The graph shown here has 4 vertices and 4 edges. And two edges of this graph have the same weight values as the remaining two. Thus, the possible spanning trees for this graph will be more than one with the exact cost.

Minimum_Spanning_Tree_Property.

Stand Out From Your Peers this Appraisal Season

Start Learning With Our FREE CoursesEnroll Now
Stand Out From Your Peers this Appraisal Season

Applications of Spanning Tree

The following are the applications of the spanning trees:

  • Telecommunication Network Building: If we want to develop a telecommunication network for the entire city, a basic naive approach will be more expensive. We can create a communications system at a lower cost by using the Minimum Spanning Tree technique. The image given below explains the difference between Naive and MST routing.

Naive_vs_MST_Routing-Spanning_trees

  • Constructing Highways or Railroads: For constructing highways or railroads the Minimum Spanning Tree approach is utilized everywhere. Given the possible routes between two cities, MST technique provides you with the optimal route. Basically, the algorithm treats the cities as the vertices and paths joining them as edges to create a subtree that will make the route fully connected and cost efficient.

Roadway_Construction_Illustration.j

  • Image Segmentation: During picture segmentation, a spanning tree is utilized to construct tiles of comparable pixels. The pixels which seem closer to each other and have the same color type are grouped together. This technique is used in every aspect of computer vision in machine learning.

image_segmentation-Spanning_Tree

Advance your career as a MEAN stack developer with the Full Stack Web Developer - MEAN Stack Master's Program. Enroll now!

Conclusion

In this tutorial, we explored Spanning Tree in a data structure. We discussed various properties of spanning trees and learned how to create these trees for a given graph topology. Later, in this tutorial, we also learned about the applications of spanning trees in order to comprehend its significance.

If you are looking for more extensive learning that goes beyond data structures and covers the principles of interactive application development, Simplilearn’s Software Development Courses will prove to be precisely suitable for you. The courses listed in the catalog above will assist you in mastering the craft of software development and prepare you for employment. Explore now and get started!

Have any questions about this article? If yes, please leave them in the comments section at the bottom of this page; we will respond to them soon!

About the Author

SimplilearnSimplilearn

Simplilearn is one of the world’s leading providers of online training for Digital Marketing, Cloud Computing, Project Management, Data Science, IT, Software Development, and many other emerging technologies.

View More
  • Disclaimer
  • PMP, PMI, PMBOK, CAPM, PgMP, PfMP, ACP, PBA, RMP, SP, and OPM3 are registered marks of the Project Management Institute, Inc.