GATE CS 2007 Q43 — Trees

To find the value of $n$, we can establish a relationship between the number of internal nodes ($I$), the number of leaves ($L$), and the degree of the tree ($n$).

Let $N$ be the total number of nodes in the $n$-ary tree. By definition, we have:

In a complete $n$-ary tree, every internal node has exactly $n$ children, and the root is the only node that is not a child of any node. Therefore, the total number of nodes can also be expressed in terms of the children of internal nodes:

Equating the two expressions for $N$:

Rearranging the equation to solve for $L$:

Given that $L = 41$ and $I = 10$, we substitute these values into the equation:

Thus, the value of $n$ is $5$, which corresponds to option C.