GATE CSE 1993 | Question: 16

Question

Prove by the principal of mathematical induction that for any binary tree, in which every non-leaf node has $2$-descendants, the number of leaves in the tree is one more than the number of non-leaf nodes.

Answer 1

Base Case :- When we have just root then, there are no non leaf nodes. So No of leaves $= 1$, No of non leaf nodes is $= 0$. Base case holds.

Induction Hypothesis :- Assume that now for $k$ internal nodes we will have $k+1$ leaves.

Inducting on no of leaves, Now we add $2$ more leaves to this tree. One of $k+1$ leaf will become internal node. So now we will have $k+1$ internal node. No of leafs will be $K+ 1 - 1$ ($1$ leaf just became internal node) $+ 2$(New leafs) . So we proved that for any binary tree, in which every non-leaf node has $2$-descendants, the number of leaves in the tree is one more than the number of non-leaf nodes.

Comments

2-descendants mean ?

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@set2018

Descendant: A node reachable by repeated proceeding from parent to child. So for every non-leaf node, we must be having two nodes which can be obtained by proceeding from parent to child.

Ref: https://en.wikipedia.org/wiki/Tree_(data_structure)#Terminology_used_in_trees

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A descendant of a node N is either N itself, a child of N, a child of a child of N and so on for any number of levels .. We say node N is an ancestor of node M if M is a descendant of N ...

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@Puja Mishra..."A descendant of a node N is either N itself, ......"

Can we call a node a descendant of itself?

Answer 2

**descendents mean immediate child or no of node from that node to leaf ??

if immediate then :
let total n nodes are in binary tree.
2*non_leaf node  + 1 = total node
non leaf node = (total node -1)/2 = (n-1)/2
no of leaf node + no of nonleaf node = total node
no of leaf node + (n-1)/2 = n
no of leaf node = (n+1)/2
                            =  (n-1 + 2)/2
                            = (n - 1)/2 + 1
                            = no of non leaf node + 1

Comments

Descendant: A node reachable by repeated proceeding from parent to child..

Ref: https://en.wikipedia.org/wiki/Tree_(data_structure)#Terminology_used_in_trees

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Sir ,how this come "total node= 2*non_leaf node  + 1  "