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Minimum number of nodes in an AVL Tree with given height
• Difficulty Level : Easy
• Last Updated : 19 Aug, 2019

Given the height of an AVL tree ‘h’, the task is to find the minimum number of nodes the tree can have.

Examples :

```Input : H = 0
Output : N = 1
Only '1' node is possible if the height
of the tree is '0' which is the root node.

Input : H = 3
Output : N = 7
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Recursive Approach : In an AVL tree, we have to maintain the height balance property, i.e. difference in the height of the left and the right subtrees can not be other than -1, 0 or 1 for each node.
We will try to create a recurrence relation to find minimum number of nodes for a given height, n(h).

• For height = 0, we can only have a single node in an AVL tree, i.e. n(0) = 1
• For height = 1, we can have a minimum of two nodes in an AVL tree, i.e. n(1) = 2
• Now for any height ‘h’, root will have two subtrees (left and right). Out of which one has to be of height h-1 and other of h-2. [root node excluded]
• So, n(h) = 1 + n(h-1) + n(h-2) is the required recurrence relation for h>=2 [1 is added for the root node]

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``using` `namespace` `std;`` ` `// Function to find``// minimum number of nodes``int` `AVLnodes(``int` `height)``{``    ``// Base Conditions``    ``if` `(height == 0)``        ``return` `1;``    ``else` `if` `(height == 1)``        ``return` `2;`` ` `    ``// Recursive function call``    ``// for the recurrence relation``    ``return` `(1 + AVLnodes(height - 1) + AVLnodes(height - 2));``}`` ` `// Driver Code``int` `main()``{``    ``int` `H = 3;``    ``cout << AVLnodes(H) << endl;``}`

## Java

 `// Java implementation of the approach`` ` `class` `GFG{``     ` ` ` `// Function to find``// minimum number of nodes``static` `int` `AVLnodes(``int` `height)``{``    ``// Base Conditions``    ``if` `(height == ``0``)``        ``return` `1``;``    ``else` `if` `(height == ``1``)``        ``return` `2``;``  ` `    ``// Recursive function call``    ``// for the recurrence relation``    ``return` `(``1` `+ AVLnodes(height - ``1``) + AVLnodes(height - ``2``));``}``  ` `// Driver Code``public` `static` `void` `main(String args[])``{``    ``int` `H = ``3``;``    ``System.out.println(AVLnodes(H));``}``}`

## Python3

 `# Python 3 implementation of the approach`` ` `# Function to find minimum ``# number of nodes``def` `AVLnodes(height):``     ` `    ``# Base Conditions``    ``if` `(height ``=``=` `0``):``        ``return` `1``    ``elif` `(height ``=``=` `1``):``        ``return` `2`` ` `    ``# Recursive function call``    ``# for the recurrence relation``    ``return` `(``1` `+` `AVLnodes(height ``-` `1``) ``+` `                ``AVLnodes(height ``-` `2``))`` ` `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``    ``H ``=` `3``    ``print``(AVLnodes(H))``     ` `# This code is contributed by``# Surendra_Gangwar`

## C#

 `// C# implementation of the approach``using` `System;`` ` `class` `GFG``{``     ` `// Function to find``// minimum number of nodes``static` `int` `AVLnodes(``int` `height)``{``    ``// Base Conditions``    ``if` `(height == 0)``        ``return` `1;``    ``else` `if` `(height == 1)``        ``return` `2;`` ` `    ``// Recursive function call``    ``// for the recurrence relation``    ``return` `(1 + AVLnodes(height - 1) + ``                ``AVLnodes(height - 2));``}`` ` `// Driver Code``public` `static` `void` `Main()``{``    ``int` `H = 3;``    ``Console.Write(AVLnodes(H));``}``}`` ` `// This code is contributed ``// by Akanksha Rai`

## PHP

 `
Output:

```7
```

Tail Recursive Approach :

• The recursive function for finding n(h) (minimum number of nodes possible in an AVL Tree with height ‘h’) is n(h) = 1 + n(h-1) + n(h-2) ; h>=2 ; n(0)=1 ; n(1)=2;
• To create a Tail Recursive Function, we will maintain 1 + n(h-1) + n(h-2) as function arguments such that rather than calculating it, we directly return its value to main function.

Below is the implementation of the above approach :

## C++

 `// C++ implementation of the approach``#include ``using` `namespace` `std;`` ` `// Function to return ``//minimum number of nodes``int` `AVLtree(``int` `H, ``int` `a = 1, ``int` `b = 2)``{``    ``// Base Conditions``    ``if` `(H == 0)``        ``return` `1;``    ``if` `(H == 1)``        ``return` `b;`` ` `    ``// Tail Recursive Call``    ``return` `AVLtree(H - 1, b, a + b + 1);``}`` ` `// Driver Code``int` `main()``{``    ``int` `H = 5;``    ``int` `answer = AVLtree(H);`` ` `    ``// Output the result``    ``cout << ``"n("` `<< H << ``") = "``         ``<< answer << endl;``    ``return` `0;``}`

## Java

 `// Java implementation of the approach``class` `GFG``{`` ` `// Function to return ``//minimum number of nodes``static` `int` `AVLtree(``int` `H, ``int` `a, ``int` `b)``{``    ``// Base Conditions``    ``if` `(H == ``0``)``        ``return` `1``;``    ``if` `(H == ``1``)``        ``return` `b;`` ` `    ``// Tail Recursive Call``    ``return` `AVLtree(H - ``1``, b, a + b + ``1``);``}`` ` `// Driver Code``public` `static` `void` `main(String[] args)``{``    ``int` `H = ``5``;``    ``int` `answer = AVLtree(H, ``1``, ``2``);`` ` `    ``// Output the result``    ``System.out.println(``"n("` `+ H + ``") = "` `+ answer);``}``}`` ` `// This code is contributed by PrinciRaj1992`

## Python3

 `# Python3 implementation of the approach`` ` `# Function to return ``# minimum number of nodes``def` `AVLtree(H, a, b):``     ` `    ``# Base Conditions``    ``if``(H ``=``=` `0``):``        ``return` `1``;``    ``if``(H ``=``=` `1``):``        ``return` `b;`` ` `    ``# Tail Recursive Call``    ``return` `AVLtree(H ``-` `1``, b, a ``+` `b ``+` `1``);`` ` `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``    ``H ``=` `5``;``    ``answer ``=` `AVLtree(H, ``1``, ``2``);`` ` `    ``# Output the result``    ``print``(``"n("``, H , ``") = "``\``        ``, answer);`` ` `# This code is contributed by 29AjayKumar`

## C#

 `// C# implementation of the approach``using` `System;``     ` `class` `GFG``{`` ` `// Function to return ``//minimum number of nodes``static` `int` `AVLtree(``int` `H, ``int` `a, ``int` `b)``{``    ``// Base Conditions``    ``if` `(H == 0)``        ``return` `1;``    ``if` `(H == 1)``        ``return` `b;`` ` `    ``// Tail Recursive Call``    ``return` `AVLtree(H - 1, b, a + b + 1);``}`` ` `// Driver Code``public` `static` `void` `Main(String[] args)``{``    ``int` `H = 5;``    ``int` `answer = AVLtree(H, 1, 2);`` ` `    ``// Output the result``    ``Console.WriteLine(``"n("` `+ H + ``") = "` `+ answer);``}``}`` ` `// This code is contributed by Princi Singh`
Output:
```n(5) = 20
```

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