Given the height of an AVL tree ‘h’, the task is to find the minimum number of nodes the tree can have.
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:
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 :
# Python3 implementation of the approach
# Function to return
# minimum number of nodes
def AVLtree(H, a, b):
# Base Conditions
if(H == 0):
if(H == 1):
# 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 , “) = “\
# This code is contributed by 29AjayKumar
n(5) = 20
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