The number of ways in which the numbers 1, 2, 3, 4, 5, 6, 7 can be inserted in an empty binary search tree, such that the resulting tree has height 6, is _______ . *Note: The height of a tree with a single node is 0.*

A binary search tree with $7$ nodes has height $6$ if and only if it is completely degenerate (a single linear chain of $7$ nodes where every non-leaf node has exactly one child).

To construct such a tree by sequential insertion:

1. For the first node (root), we must choose either the minimum or the maximum among the set of available numbers $\{1, 2, 3, 4, 5, 6, 7\}$. If any intermediate number is chosen, it would force at least one element into its left subtree and at least one into its right subtree, preventing the tree from being a single chain of length 6. Thus, there are 2 choices (either $1$ or $7$).

2. After choosing the root, $6$ numbers remain. To maintain a single chain, the second node inserted must again be either the minimum or maximum of the remaining elements: 2 choices.

3. Similarly, at each step $k$ for $k = 1, 2, \dots, 6$, there are $2$ choices (either the smallest or largest of the remaining elements).

4. For the 7th node, only $1$ element remains: 1 choice.

Total number of insertion sequences:

Thus, the correct answer is 64.