# Basic CS-Data Structures [TCS Placement]: Sample Questions 3 - 4 of 28

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## Question 3

Data Structures

### Explanation

- Tuple is a fixed-size collection that can have elements of either same or different data types.
- Similar to arrays, a user must specify the size of a tuple at the time of declaration.
- Tuples are allowed to hold 1 to 8 elements and if there are more than 8 elements, then the 8
^{th}element can be defined as another tuple. - Tuples can be specified as parameter or return type of a method.

## Question 4

Data Structures

### 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 n
^{n-2}number of spanning trees, where n is the number of nodes. - In the above addressed example, 3
^{3 − 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