# Tree DS

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1. How many Binary trees can be made with:

(a) 3 unlabelled nodes?

(b) 3 labelled nodes?

2. How many Binary Search trees can be made with:

(a) 3 unlabelled nodes?

(b) 3 labelled nodes?

3. How many AVL trees can be made with:

(a) 3 unlabelled nodes?

(b) 3 labelled nodes?

Can these be generalised for 'n' nodes?

in DS

How many Binary trees can be made with :-

1) Un- labeled :- There is a formula called CATALAN NUMBER = $\frac{\binom{2n}{n}}{n+1}$

Called them as patterns

2) labeled :- for each pattern above, there are n! trees ===> total = $\frac{\binom{2n}{n}}{n+1} * n!$

How many Binary Search trees can be made with :-

1) Un- labeled :- There is a formula called CATALOG NUMBER = $\frac{\binom{2n}{n}}{n+1}$

Called them as patterns

2) labeled :- for each pattern above, there only 1 way to labeled the nodes of the tree ===> total = $\frac{\binom{2n}{n}}{n+1}$

How many AVL trees can be made with :-

1) with 3 un-labeled nodes:- only one way ( i don't know general formula for un-labeled )

2) labeled :- for each pattern above, there only 1 way to labeled the nodes of the tree ===> total = No.of trees with un-labeled.

edited
2
Minor correction -- the number you are referring to you is called the Catalan number.
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How can a binary search tree be unlabelled?

We can form a Bst only when it is labelled with values.

Without labels it is just a binary tree..
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@Human

yes,you are absolutely right.... without labeling you can't decide it is Binary tree or Binary search tree

But why i wrote unlabeled trees is " First we have to take a pattern and Fix it, then it lead to unique BST"

For simple calculation purpose only, i used it

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Ok brother.

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