The height of a binary search tree is defined as the number of edges in the longest path from the root to a leaf node. In this question, we are given the numbers 1, 2, 3, 4, 5, 6, and 7 and we need to insert them into an empty binary search tree such that the resulting tree has a height of 6.

Let's analyze the problem step by step:

1. Root Node:
- The root node can be any of the numbers 1 to 7.
- So, there are 7 possible choices for the root node.

2. Level 1:
- The numbers that are less than the root node will be placed in the left subtree, and the numbers that are greater than the root node will be placed in the right subtree.
- As we want the resulting tree to have a height of 6, the root node will have 6 levels below it.
- Since the root node is fixed, there are no choices to make at this level.

3. Level 2 to Level 6:
- At each level, the numbers that are less than the parent node will go in the left subtree, and the numbers that are greater than the parent node will go in the right subtree.
- The number of choices at each level depends on the number of nodes already placed in the tree.
- At level 2, there will be 1 node in the tree (the root node), so there are 6 choices for the second level.
- At level 3, there will be 2 nodes in the tree, so there are 5 choices for the third level.
- Similarly, at level 4, there are 4 choices; at level 5, there are 3 choices; and at level 6, there are 2 choices.

4. Total Number of Combinations:
- To find the total number of combinations, we multiply the number of choices at each level.
- So, the total number of combinations is 7 * 6 * 5 * 4 * 3 * 2 = 5040.

However, we need to consider that the binary search tree is unique only up to its structure, not the values it contains. In other words, two different binary search trees can have the same structure but different values. Therefore, we need to divide the total number of combinations by the number of possible ways to arrange the 7 numbers in a tree of height 6.

5. Number of Ways to Arrange the Numbers:
- To find the number of ways to arrange the numbers, we can use the formula for finding the number of permutations of a set.
- The number of permutations of n distinct objects is given by n!
- In this case, we have 7 distinct numbers, so the number of ways to arrange them is 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040.

6. Final Answer:
- To obtain the number of unique binary search trees of height 6, we divide the total number of combinations by the number of ways to arrange the numbers: 5040 / 5040 = 1.

Therefore, the correct answer is option 'C' - 64.