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The number of BST's possible with $6$ nodes numbered $1$,$2$,$3$,$4$,$5$ and $6$ with exactly one leaf node are .......................

OR

The number of BST's possible with $6$ nodes numbered $1$,$2$,$3$,$4$,$5$ and $6$ having a height of $5$ are .................…

( note :- height of a root is 0 )
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Best answer
17 votes
17 votes

The ans is 25=32

Here the elements are  1,2,3,4,5,6 and we need only 1 leaf 

1.So let us fix a root node: To fix the root node we have 2 choice either 1 or 6 from  1,2,3,4,5,6

(if we select any non-extreme elements then only 1 leaf is not possible)

Let us say we select 1 as root node 

2.In each and every level we have 2 different option that is to select i,e; either of extreme elements

So in 2nd level we can select 2,3,4,5,6

Let us say we select 6 in 2nd level 

3.We still have 2 option to select in 2,3,4,5

So this is true in all levels except last level since only 1 element is left.

Therefore, total possibilities are : 2 * 2 * 2 * 2 = 24

The same condition is true for 6 as a root

Hence, 24* 2 (1 as root or 6 as root)  =  25

edited by
7 votes
7 votes

I'm getting 32. But I followed brute force method. 16 rooted on node 1 and 16 rooted on node 6.

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