- Data Structures & Algorithms
- DSA - Home
- DSA - Overview
- DSA - Environment Setup
- DSA - Algorithms Basics
- DSA - Asymptotic Analysis
- Data Structures
- DSA - Data Structure Basics
- DSA - Data Structures and Types
- DSA - Array Data Structure
- Linked Lists
- DSA - Linked List Data Structure
- DSA - Doubly Linked List Data Structure
- DSA - Circular Linked List Data Structure
- Stack & Queue
- DSA - Stack Data Structure
- DSA - Expression Parsing
- DSA - Queue Data Structure
- Searching Algorithms
- DSA - Searching Algorithms
- DSA - Linear Search Algorithm
- DSA - Binary Search Algorithm
- DSA - Interpolation Search
- DSA - Jump Search Algorithm
- DSA - Exponential Search
- DSA - Fibonacci Search
- DSA - Sublist Search
- DSA - Hash Table
- Sorting Algorithms
- DSA - Sorting Algorithms
- DSA - Bubble Sort Algorithm
- DSA - Insertion Sort Algorithm
- DSA - Selection Sort Algorithm
- DSA - Merge Sort Algorithm
- DSA - Shell Sort Algorithm
- DSA - Heap Sort
- DSA - Bucket Sort Algorithm
- DSA - Counting Sort Algorithm
- DSA - Radix Sort Algorithm
- DSA - Quick Sort Algorithm
- Graph Data Structure
- DSA - Graph Data Structure
- DSA - Depth First Traversal
- DSA - Breadth First Traversal
- DSA - Spanning Tree
- Tree Data Structure
- DSA - Tree Data Structure
- DSA - Tree Traversal
- DSA - Binary Search Tree
- DSA - AVL Tree
- DSA - Red Black Trees
- DSA - B Trees
- DSA - B+ Trees
- DSA - Splay Trees
- DSA - Tries
- DSA - Heap Data Structure
- Recursion
- DSA - Recursion Algorithms
- DSA - Tower of Hanoi Using Recursion
- DSA - Fibonacci Series Using Recursion
- Divide and Conquer
- DSA - Divide and Conquer
- DSA - Max-Min Problem
- DSA - Strassen's Matrix Multiplication
- DSA - Karatsuba Algorithm
- Greedy Algorithms
- DSA - Greedy Algorithms
- DSA - Travelling Salesman Problem (Greedy Approach)
- DSA - Prim's Minimal Spanning Tree
- DSA - Kruskal's Minimal Spanning Tree
- DSA - Dijkstra's Shortest Path Algorithm
- DSA - Map Colouring Algorithm
- DSA - Fractional Knapsack Problem
- DSA - Job Sequencing with Deadline
- DSA - Optimal Merge Pattern Algorithm
- Dynamic Programming
- DSA - Dynamic Programming
- DSA - Matrix Chain Multiplication
- DSA - Floyd Warshall Algorithm
- DSA - 0-1 Knapsack Problem
- DSA - Longest Common Subsequence Algorithm
- DSA - Travelling Salesman Problem (Dynamic Approach)
- Approximation Algorithms
- DSA - Approximation Algorithms
- DSA - Vertex Cover Algorithm
- DSA - Set Cover Problem
- DSA - Travelling Salesman Problem (Approximation Approach)
- Randomized Algorithms
- DSA - Randomized Algorithms
- DSA - Randomized Quick Sort Algorithm
- DSA - Karger’s Minimum Cut Algorithm
- DSA - Fisher-Yates Shuffle Algorithm
- DSA Useful Resources
- DSA - Questions and Answers
- DSA - Quick Guide
- DSA - Useful Resources
- DSA - Discussion
Spanning Tree
What is Spanning Tree?
A spanning tree is a subset of Graph G, which has all the vertices covered with minimum possible number of edges. Hence, a spanning tree does not have cycles and it cannot be disconnected..
By this definition, we can draw a conclusion that every connected and undirected Graph G has at least one spanning tree. A disconnected graph does not have any spanning tree, as it cannot be spanned to all its vertices.
We found three spanning trees off one complete graph. A complete undirected graph can have maximum nn-2 number of spanning trees, where n is the number of nodes. In the above addressed example, n is 3, hence 33−2 = 3 spanning trees are possible.
General Properties of Spanning Tree
We now understand that one graph can have more than one spanning tree. Following are a few properties of the spanning tree connected to graph G −
A connected graph G can have more than one spanning tree.
All possible spanning trees of graph G, have the same number of edges and vertices.
The spanning tree does not have any cycle (loops).
Removing one edge from the spanning tree will make the graph disconnected, i.e. the spanning tree is minimally connected.
Adding one edge to the spanning tree will create a circuit or loop, i.e. the spanning tree is maximally acyclic.
Mathematical Properties of Spanning Tree
Spanning tree has n-1 edges, where n is the number of nodes (vertices).
From a complete graph, by removing maximum e - n + 1 edges, we can construct a spanning tree.
A complete graph can have maximum nn-2 number of spanning trees.
Thus, we can conclude that spanning trees are a subset of connected Graph G and disconnected graphs do not have spanning tree.
Application of Spanning Tree
Spanning tree is basically used to find a minimum path to connect all nodes in a graph. Common application of spanning trees are −
Civil Network Planning
Computer Network Routing Protocol
Cluster Analysis
Let us understand this through a small example. Consider, city network as a huge graph and now plans to deploy telephone lines in such a way that in minimum lines we can connect to all city nodes. This is where the spanning tree comes into picture.
Minimum Spanning Tree (MST)
In a weighted graph, a minimum spanning tree is a spanning tree that has minimum weight than all other spanning trees of the same graph. In real-world situations, this weight can be measured as distance, congestion, traffic load or any arbitrary value denoted to the edges.
Minimum Spanning-Tree Algorithm
We shall learn about two most important spanning tree algorithms here −
These two algorithms are Greedy algorithms.