Data Structures | Binary Trees | Question 15

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)


Answer: (C)

Explanation: 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.

Let us see following for example

             o
        /    |    \
      o      o      o
   / | \          / | \
  o  o  o        o  o  o
                  / | \
                 o  o  o

k = 3
Number of internal nodes n = 4
Number of leaf nodes = (k-1)*n  + 1
                     = (3-1)*4 + 1
                     = 9 


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