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Minimum number of nodes in an AVL Tree with given height

  • Difficulty Level : Easy
  • Last Updated : 18 Jun, 2021

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

Examples : 

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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

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 <bits/stdc++.h>
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




# Python3 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




<?php
// PHP implementation of the approach
 
// Function to find minimum
// number of nodes
function AVLnodes($height)
{
    // Base Conditions
    if ($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
$H = 3;
echo(AVLnodes($H));
 
// This code is contributed
// by Code_Mech.

Javascript




<script>
 
// Javascript implementation of the approach
 
// Function to find
// minimum number of nodes
function AVLnodes(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
let H = 3;
 
document.write(AVLnodes(H));
 
// This code is contributed by decode2207
 
</script>
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 <iostream>
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

Javascript




<script>
    // Javascript implementation of the approach
     
    // Function to return
    //minimum number of nodes
    function 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);
    }
     
    let H = 5;
    let answer = AVLtree(H, 1, 2);
   
    // Output the result
    document.write("n(" + H + ") = " + answer);
 
// This code is contributed by mukesh07.
</script>
Output: 
n(5) = 20

 




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