Table of contents  
Spanning Tree and Minimum Spanning Tree 
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In this tutorial, you will learn about spanning tree and minimum spanning tree with help of examples and figures.
Before we learn about spanning trees, we need to understand two graphs: undirected graphs and connected graphs.
An undirected graph is a graph in which the edges do not point in any direction (ie. the edges are bidirectional).
Undirected Graph
A connected graph is a graph in which there is always a path from a vertex to any other vertex.
Connected Graph
A spanning tree is a subgraph of an undirected connected graph, which includes all the vertices of the graph with a minimum possible number of edges. If a vertex is missed, then it is not a spanning tree.
The edges may or may not have weights assigned to them.
The total number of spanning trees with n vertices that can be created from a complete graph is equal to n^{(n2)}.
If we have n = 4, the maximum number of possible spanning trees is equal to 4^{42} = 16. Thus, 16 spanning trees can be formed from a complete graph with 4 vertices.
Example of a Spanning Tree
Let's understand the spanning tree with examples below:
Let the original graph be:
Normal graphSome of the possible spanning trees that can be created from the above graph are:
A spanning tree
Minimum Spanning Tree
A minimum spanning tree is a spanning tree in which the sum of the weight of the edges is as minimum as possible.
Example of a Spanning Tree
Let's understand the above definition with the help of the example below.
The initial graph is:
Weighted graphThe possible spanning trees from the above graph are:
Minimum spanning tree  1
Minimum spanning tree  2
Minimum spanning tree  3
Minimum spanning tree  4
The minimum spanning tree from the above spanning trees is:
Minimum spanning tree
The minimum spanning tree from a graph is found using the following algorithms:
Spanning Tree Applications
Minimum Spanning tree Applications
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