GATE-CS-2007 Question 5

This is a question from the GATE-CS-2007 exam. The question is focused on the understanding of tree data structure.

Question

Suppose we have a binary tree with n nodes. The height of the tree is h. What is the minimum and maximum number of edges in the tree?

Solution

The minimum number of edges in a binary tree with n nodes is n-1. This is achieved when the tree is a linear chain with all nodes connected in a straight line. Since the tree has n nodes, there are n-1 edges connecting them.

The maximum number of edges in a binary tree with n nodes is n-1. This is achieved when the tree is a complete binary tree. A complete binary tree is a binary tree in which all levels are completely filled, except possibly the last level, which is filled from left to right.

A complete binary tree with n nodes has a height of log(n) base 2. Therefore, the maximum number of nodes in a complete binary tree with height h is 2^(h+1)-1. This means that the maximum number of edges in a complete binary tree with n nodes is 2^(h+1)-2.

Therefore, the minimum and maximum number of edges in a binary tree with n nodes are n-1 and 2^(h+1)-2, respectively.

Conclusion

To summarize, the minimum and maximum number of edges in a binary tree with n nodes are n-1 and 2^(h+1)-2, respectively. Understanding these properties of the binary tree is important when working with tree data structures in computer science.