# Spanning Tree and Minimum Spanning Tree

#### 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).

A **connected graph** is a graph in which there is always a path from a vertex to any other vertex.

## Spanning tree

A spanning tree is a sub-graph 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`

.^{(n-2)}

If we have `n = 4`

, the maximum number of possible spanning trees is equal to `4`

^{4-2}`= 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:

Some of the possible spanning trees that can be created from the above graph are:

## 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:

The possible spanning trees from the above graph are:

The minimum spanning tree from the above spanning trees is:

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

- Prim’s Algorithm
- Kruskal’s Algorithm

## Spanning Tree Applications

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

## Minimum Spanning tree Applications

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

# Python Example for Beginners

## Two Machine Learning Fields

There are two sides to machine learning:

**Practical Machine Learning:**This is about querying databases, cleaning data, writing scripts to transform data and gluing algorithm and libraries together and writing custom code to squeeze reliable answers from data to satisfy difficult and ill defined questions. It’s the mess of reality.**Theoretical Machine Learning**: This is about math and abstraction and idealized scenarios and limits and beauty and informing what is possible. It is a whole lot neater and cleaner and removed from the mess of reality.

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