# Minimum number of nodes in an AVL Tree with given height

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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