# GATE | GATE-CS-2000 | Question 14

Consider the following nested representation of binary trees: (X Y Z) indicates Y and Z are the left and right sub stress, respectively, of node X. Note that Y and Z may be NULL, or further nested. Which of the following represents a valid binary tree?**(A)** (1 2 (4 5 6 7))

**(B)** (1 (2 3 4) 5 6) 7)**(C)** (1 (2 3 4)(5 6 7))**(D)** (1 (2 3 NULL) (4 5))**Answer:** **(C)****Explanation:** To solve this question we have to look at two things:

1) In a binary tree, a node may have at most 2 children.

2) To construct binary tree from the given sequences above, innermost parenthesis should be worked first.

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In Option A : ( 4 5 6 7 ) is there, which says that node 4 has got three children, which is wrong for a binary tree, and also in the question, only ( X Y Z ) is defined, i.e. a node X can have at most 2 children, which will be the roots of subtrees Y and Z.

In Option B : after working on innermost (2 3 4), where 2 is a node of the binary tree, 3 is left subtree of node 2 and 4 is right subtree of node 2. From this we get (1 2 5 6). Here 2 has come from the root of subtree ( 2 3 4 ). Now again we don’t have any definition for ( 1 2 5 6). Hence invalid.

In Option C: after working on ( 2 3 4) and ( 5 6 7 ) we get ( 1 2 5 ) where 2 has come from the root of subtree ( 2 3 4 ) and 5 has come from the root of subtree ( 5 6 7 ). Now, in ( 1 2 5 ) node 1 is the root of the binary tree, and subtree with root 2 is the left subtree and subtree with root 5 is the right subtree at root node 1. Hence it is giving valid binary tree.

In Option D: It is given as ( 2 3 NULL), here N,U,L and L are given as different elements, which is again wrong as according to ( X Y Z) definition, a node can have at most 2 children.

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