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In a binary tree with n nodes, every node has an odd number of descendants. Every node is considered to be its own descendant. What is the number of nodes in the tree that have exactly one child?
  • a)
    0
  • b)
    1
  • c)
    (n-1)/2
  • d)
    n-1
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
In a binary tree with n nodes, every node has an odd number of descend...
It is mentioned that each node has odd number of descendants including node itself, so all nodes must have even number of descendants 0, 2, 4 so on. Which means each node should have either 0 or 2 children. So there will be no node with 1 child. Hence 0 is answer. Following are few examples.
Such a binary tree is full binary tree (a binary tree where every node has 0 or 2 children).
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Most Upvoted Answer
In a binary tree with n nodes, every node has an odd number of descend...
Explanation:

  • Let's assume there are k nodes with exactly one child in the binary tree.

  • Since every node has an odd number of descendants, every node must have an odd number of children (1 or 3).

  • Thus, every node that is not a leaf must have 2 children.

  • Therefore, the number of leaves in the tree is (n-k).

  • Since every non-leaf node has 2 children, the total number of children in the tree is 2(n-k).

  • However, every node is counted twice as a child (once for each of its parents), so we must divide by 2 to get the total number of nodes in the tree, which is n-k.

  • Since every node has an odd number of children, the total number of children must be odd.

  • Therefore, n-k must be odd.

  • However, if k is greater than 0, then every node with exactly one child is not a leaf, so the total number of leaves in the tree is less than n-k.

  • This is a contradiction, since the total number of leaves in the tree must be (n-k).

  • Therefore, k must be 0.

  • Thus, there are no nodes in the tree with exactly one child.


Therefore, the correct answer is option A.
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