What is the minimum number of nodes in an AVL tree of height 7?

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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

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:

#include <bits/stdc++.h>
using namespace std;
int AVLnodes(int height)
{
    if (height == 0)
        return 1;
    else if (height == 1)
        return 2;
    return (1 + AVLnodes(height - 1) + AVLnodes(height - 2));
}
int main()
{
    int H = 3;
    cout << AVLnodes(H) << endl;
}

class GFG{
static int AVLnodes(int height)
{
    if (height == 0)
        return 1;
    else if (height == 1)
        return 2;
    return (1 + AVLnodes(height - 1) + AVLnodes(height - 2));
}
public static void main(String args[])
{
    int H = 3;
    System.out.println(AVLnodes(H));
}
}

def AVLnodes(height):
    if (height == 0):
        return 1
    elif (height == 1):
        return 2
    return (1 + AVLnodes(height - 1) +
                AVLnodes(height - 2))
if __name__ == '__main__':
    H = 3
    print(AVLnodes(H))

using System;
class GFG
{
static int AVLnodes(int height)
{
    if (height == 0)
        return 1;
    else if (height == 1)
        return 2;
    return (1 + AVLnodes(height - 1) +
                AVLnodes(height - 2));
}
public static void Main()
{
    int H = 3;
    Console.Write(AVLnodes(H));
}
}

<?php

function AVLnodes($height)

{

if ($height == 0)

return 1;

else if ($height == 1)

return 2;

return (1 + AVLnodes($height - 1) +

AVLnodes($height - 2));

}

$H = 3;

echo(AVLnodes($H));

<script>
function AVLnodes(height)
{
    if (height == 0)
        return 1;
    else if (height == 1)
        return 2;
    return (1 + AVLnodes(height - 1) +
                AVLnodes(height - 2));
}
let H = 3;
document.write(AVLnodes(H));
</script>

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 :

#include <iostream>
using namespace std;
int AVLtree(int H, int a = 1, int b = 2)
{
    if (H == 0)
        return 1;
    if (H == 1)
        return b;
    return AVLtree(H - 1, b, a + b + 1);
}
int main()



{
    int H = 5;
    int answer = AVLtree(H);
    cout << "n(" << H << ") = "
         << answer << endl;
    return 0;
}

class GFG
{
static int AVLtree(int H, int a, int b)
{
    if (H == 0)
        return 1;
    if (H == 1)
        return b;
    return AVLtree(H - 1, b, a + b + 1);
}
public static void main(String[] args)
{
    int H = 5;
    int answer = AVLtree(H, 1, 2);
    System.out.println("n(" + H + ") = " + answer);
}
}

def AVLtree(H, a, b):

if(H == 0):

return 1;

if(H == 1):

return b;

return AVLtree(H - 1, b, a + b + 1);

if __name__ == '__main__':

H = 5;

answer = AVLtree(H, 1, 2);

print("n(", H , ") = "\

, answer);

using System;
class GFG
{
static int AVLtree(int H, int a, int b)
{
    if (H == 0)
        return 1;
    if (H == 1)
        return b;
    return AVLtree(H - 1, b, a + b + 1);
}
public static void Main(String[] args)
{
    int H = 5;
    int answer = AVLtree(H, 1, 2);
    Console.WriteLine("n(" + H + ") = " + answer);
}
}

<script>
    function AVLtree(H, a, b)
    {
        if (H == 0)
            return 1;
        if (H == 1)
            return b;
        return AVLtree(H - 1, b, a + b + 1);
    }
    let H = 5;
    let answer = AVLtree(H, 1, 2);
    document.write("n(" + H + ") = " + answer);
</script>

Q&A What

What is the minimum number of nodes in an AVL tree of height 7?

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