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Q. The degree of a node is defined as the number of children it has. Shear show that in any binary tree, the total number of leaves is one more than the number of nodes of degree 2
Ans:
Let n be the total number of nodes of degree 2 in a binary tree T. We have to show that the number of leaves in T is n+1. Let us prove it by induction technique.
For n=1, the result is certain that the leaves are two i.e. 1+1.
Let the formulae be true for n=k, i.e. if there are n nodes of degree 2 in T then T has k+1 leaves of it.
If we add the two children to one of the leaf node then, the number nodes with degree two will be k+1 and number of leaf node will be (k+1)-1+2 = (k+1)+1. Hence we can conclude by induction method that if a binary tree t has n nodes of degree 2 It then it has n+1 leaves.
Handout 15 COMP 264: Introduction to Computer Systems (Section 001) Spring 2013 R. I. Greenberg Computer Science Department Loyola University Water TowerCampus, Lewis Towers 524 82
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