We are given a set of $n$ distinct elements and an unlabeled binary tree with $n$ nodes. In how many ways can we populate the tree with the given set so that it becomes a binary search tree?

In a Binary Search Tree (BST), an in-order traversal of the tree visits the keys in strictly increasing sorted order.

• Since all $n$ elements in the given set are distinct, there is exactly one sorted sequence of these elements.

• For a given fixed binary tree structure with $n$ nodes, the in-order traversal visits the nodes in a uniquely determined order of $n$ positions.

• To satisfy the binary search tree property, the $k$-th smallest element must be placed at the $k$-th node visited during the in-order traversal for each $k \in \{1, 2, \dots, n\}$.

Therefore, there is exactly 1 way to place the elements into the fixed binary tree structure such that it becomes a valid binary search tree.

Hence, the correct option is (B) [p1].