# TCS Placement: Sample Questions 61 - 61 of 502

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## Question 61

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### Describe in Detail

Essay▾

What is a spanning tree?

### Explanation

• A spanning tree is associated with a network where all the nodes of the graph appear on the tree once.
• A minimum spanning tree is a spanning tree organized so that the total edge weight between nodes is minimized.
• The cost of the spanning tree is the sum of the weights of all the edges in the tree. There can be many spanning trees of a graph.
• Minimum spanning tree is the spanning tree where the cost is minimum among all the spanning trees. There also can be many minimum spanning trees.
• Minimum spanning tree has direct application in the design of networks. It is used in algorithms approximating the travelling salesman problem, multi-terminal minimum cut problem and minimum-cost weighted perfect matching.
• Spanning trees are a subset of connected Graph G and disconnected graphs do not have spanning tree.
• Thus, a spanning tree is a subset of Graph G, which has all the vertices covered with minimum possible number of edges- a spanning tree does not have cycles and it cannot be disconnected.
• Every connected and undirected Graph G thus has at least one spanning tree but a disconnected graph does not have any spanning tree, as it cannot be spanned to all its vertices.
• In the figure above, there are three spanning trees or one complete graph.

## Properties of Spanning Tree

• 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, 33 − 2 = 3 spanning trees are possible.
• From a complete graph, by removing maximum e - n + 1 edges, we can construct a spanning tree.
• Spanning tree with n nodes has n-1 edges

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