How many permutations of a given array result in BST's of height 2?

Question

A BST is generated (by successive insertion of nodes) from each permutation of keys from the set {1,2,3,4,5,6,7}. How many permutations determine trees of height two?

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Answer 1

Consider how the tree would be height 2?

-It needs to have 4 as root, 2 as the left child, 6 right child, etc.

How come 4 is the root?

-It needs to be the first inserted. So we have one number now, 6 still can move around in the permutation.

And?

-After the first insert there are still 6 places left, 3 for the left and 3 for the right subtrees. That's 6 choose 3 = 20 choices.

Now what?

-For the left and right subtrees, their roots need to be inserted first, then the children's order does not affect the tree - 2, 1, 3 and 2, 3, 1 gives the same tree. That's 2 for each subtree, and 2 * 2 = 4 for the left and right subtrees.

So?

In conclusion: C(6, 3) * 2 * 2 = 20 * 2 * 2 = 80.