Spanning Tree and Minimum Spanning Tree

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
Undirected Graph

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

Connected Graph
Connected Graph

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 44-2 = 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:

initial tree
Normal graph

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

spanning tree
A spanning tree
spanning tree
A spanning tree
spanning tree
A spanning tree
spanning tree
A spanning tree
spanning tree
A spanning tree
spanning tree
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:

initial graph
Weighted graph

The possible spanning trees from the above graph are:

minimum spanning tree (mst)
Minimum spanning tree - 1
minimum spanning tree (mst)
Minimum spanning tree - 2
minimum spanning tree (mst)
Minimum spanning tree - 3
minimum spanning tree (mst)
Minimum spanning tree - 4

The minimum spanning tree from the above spanning trees is:

minimum spanning tree (mst)
Minimum spanning tree

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

  1. Prim's Algorithm
  2. 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.
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