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In a complete k-ary tree, every internal node has exactly k children or no child. The number of leaves in such a tree with n internal nodes is:
  • a)
    nk
  • b)
    (n – 1) k+ 1
  • c)
    n( k – 1) + 1
  • d)
    n(k – 1)
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
In a complete k-ary tree, every internal node has exactly k children o...
 For an k-ary tree where each node has k children or no children, following relation holds
L = (k-1)*n + 1
Where L is the number of leaf nodes and n is the number of internal nodes.
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Most Upvoted Answer
In a complete k-ary tree, every internal node has exactly k children o...
Explanation:


A complete k-ary tree has every internal node with exactly k children or no child. Let's assume there are n internal nodes in the tree.

Number of children:


Since every internal node has exactly k children or no child, the number of children in the tree can be calculated as follows:

Number of children = k * n

Number of leaves:


Since a leaf node has no children, we can count the number of leaf nodes in the tree.

In a complete k-ary tree, the number of leaf nodes can be expressed as:

Number of leaves = Number of children + 1

This can be derived as follows:
- Every internal node has k children or no child.
- If an internal node has k children, then there are k leaf nodes below it.
- Therefore, the total number of leaf nodes in the tree is equal to the number of children plus 1 (for the root node).

Substituting the value of the number of children, we get:

Number of leaves = k * n + 1

Simplification:


We can simplify the above equation as follows:

Number of leaves = n * (k - 1) + 1

This can be derived as follows:
- Subtract k from both sides of the equation: Number of leaves - k * n = 1
- Factorize: Number of leaves - (k * n + 1) = -k
- Multiply by -1: (k * n + 1) - Number of leaves = k
- Substitute the value of the number of children: (k * n + 1) - (k * n) = k
- Simplify: Number of leaves = n * (k - 1) + 1

Therefore, the number of leaves in a complete k-ary tree with n internal nodes is n * (k - 1) + 1.

Hence, the correct answer is option C.
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In a complete k-ary tree, every internal node has exactly k children or no child. The number of leaves in such a tree with n internal nodes is:a)nkb)(n – 1) k+ 1c)n( k – 1) + 1d)n(k – 1)Correct answer is option 'C'. Can you explain this answer?
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