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How many distinct binary search trees can be created out of $4$ distinct keys?

1. $5$
2. $14$
3. $24$
4. $42$
edited | 1.8k views

number of distinct BSTs = $\frac{^{2n}C_n}{n+1}$ (or) =$\frac{(2n)!}{(n+1)!n!}$

For a given Binary tree structure, there can be only $1$ BST. Hence, no. of different BSTs with $n$ nodes will be equal to the no. of different binary tree structures possible for $n$ nodes.

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Yes it will be equal to Catalan no.
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sir distinct keys does not mean labelled nodes?
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no because distinct keys in BST must satisfy a property. while in a labelled nodes binary tree there is  no such restriction

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@Arjun sir

Distinct  Key  BST and Unlabeled node  BST Both are same and It is equal to Catalan Number?
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Lakshman Patel RJIT  yes try creating distinct binary trees with 3 unlabeled nodes and also try creating BST's with keys {1,2,3} and it is unlabeled Binary tree not unlabeled BST

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@Mk Utkarsh I'm little bit confuse

can you explain?

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$\text{No. of BST with n nodes}$=$\sum_{i=1}^{n} T(i-1)*T(n-i)$ $\text{where}$ $T(0)=T(1)=1$
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Distinct  Key  BST and Unlabeled node  BST Both are same and It is equal to Catalan Number?

@Lakshman Patel RJIT If you have given a particular structure of n nodes(any structure among all possible as in un labeled nodes) there is only one way to fill it to satisfy BST property. Here they are asking BST possible using 4 distinct keys. What you can do is form all possible structure of un labeled nodes(Catalan number) and then in each structure fill the keys using BST property. Hence your statement is true.

Number of binary trees with n-vertices =2n!/n!(n+1)! =14 so option B is answer

Incase you forgot the formula,

Number of BST's with 2 keys (1,2) = 2

Number of BST's with 3 keys (1,2,3) = 2 + 2 + 1 = (Root is 3) + (Root is 1)  +  (Root is 2)  = 5

Number of BST's with 4 keys (1,2,3,4) = 5 + 5 + 2 + 2 = 14

Number of BST's with 5 keys (1,2,3,4,5) = 14 + 14 + 5 + 5 + 2 * 2 = 42

Number of BST's with 5 keys (1,2,3,4,5) = Root 5 + Root 1 + Root 4 + Root 2 + Root 3 (2 trees with 2 keys on either side)

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