**Spanning Tree**: In this tutorial, you will learn about **spanning trees** and **minimum spanning trees** with the assistance of examples and figures.

Before we learn about spanning trees, we need to understand two graphs: undirected graphs and directed graphs.

An **undirected graph** is a graph wherein the edges don’t point toward any path (ie. the edges are bidirectional).

A **connected graph** is a graph wherein there is consistently a way from a vertex to some other vertex.

Contents

## What is a spanning tree?

The **spanning trees** of a graph (G) is a subset of G that covers the entirety of its vertices using the minimum number of edges.

A few properties of a spanning trees can be concluded from this definition:

- Since “a spanning tree covers the entirety of the vertices”, it can’t be disconnected.
- Spanning trees can’t have any cycles and comprise (n-1) edges (where n is the number of vertices of the graph) since “it uses the minimum number of edges”.

A graph can have more than one spanning tree. Truth be told, a total graph can have a maximum of *n** ^{n}*−

^{2}spanning trees.

## Spanning trees

A ** spanning trees** is a sub-graph of an undirected associated graph, which incorporates all the vertices of the graph with a minimum conceivable number of edges. In the event that a vertex is missed, then it is not a spanning trees.

The edges could conceivably have weights assigned to them.

The total number of spanning trees with n vertices that can be created from a total graph is equivalent to n

^{(n-2)}

On the off chance that we have n = 4, the maximum number of conceivable spanning trees is equivalent to 4^{4-2} = 16. Along these lines, 16 spanning trees can be formed from a complete graph with 4 vertices.

## Illustration of a Spanning Trees

Let’s understand the spanning trees with models underneath:

Let the original graph be:

Normal graph

A portion of the conceivable spanning trees that can be created from the above graph is:

A spanning tree

A spanning tree

A spanning tree

A spanning tree

A spanning tree

A spanning tree

## Minimum Spanning Trees

A minimum spanning trees is a spanning trees in which the sum of the weight of the edges is as minimum as possible.

## Illustration of a Spanning Trees

Let’s understand the above definition with the assistance of the example underneath.

The underlying diagram is:

Weighted graph

The 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 trees from the above spanning trees is:

Minimum spanning tree

The minimum spanning trees from a graph is found using the following algorithms:

## Spanning Trees Applications

- Computer Network Routing Protocol
- Cluster Analysis
- Civil Network Planning

## Minimum Spanning trees Applications

- To find paths in the map
- To design networks like telecommunication networks, water supply networks, and electrical grids.

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