Total number of possible Binary Search Trees and Binary Trees with n keys

Last Updated : 11 Feb, 2025

Given an integer n, the task is to find the total number of unique Binary Search trees And unique Binary trees that can be made using values from 1 to n.

Examples:

Input: n = 3
Output: BST = 5
BT = 30
Explanation: For n = 3, Total number of binary search tree will be 5 and total Binary tree will b 30.
Input: n = 5
Output: BST = 42
BT = 5040
Explanation: For n = 5, Total number of binary search tree will be 42 and total Binary tree will b 5040.

Approach:

1. Unique Binary Search Trees (BSTs):
First let's see how we find Total number of binary search tree with n nodes. A binary search tree (BST) maintains the property that elements are arranged based on their relative order. Let’s define C(n) as the number of unique BSTs that can be constructed with n nodes. This is given by the following recursive formula:
C(n) = Σ(i = 1 to n) C(i-1) * C(n-i).
This formula corresponds to the recurrence relation for the nth Catalan number. Please refer to Number of Unique BST with N Keys for better understanding and proof. We just need to find nth catalan number. First few catalan numbers are 1 1 2 5 14 42 132 429 1430 4862, … (considered from 0th number).
Formula of catalan number is (1 / n+1) * ( ^2*nCn). Please refer to Applications of Catalan Numbers.
2. Unique Binary Trees (General Binary Trees):
For general binary trees, the nodes do not have to follow the Binary Search Tree property. The total number of unique binary trees is calculated as: Total Binary Trees = countBST(n) * n!

C++

// C++ code of finding Number of Unique // BST and BT with N Keys   #include <iostream> using namespace std;  // Function to calculate the binomial // coefficient C(n, k) int binomialCoeff(int n, int k) {        	// C(n, k) is the same as C(n, n-k)     if (k > n - k) {         k = n - k;     }      int res = 1;        // Calculate the value of n! / (k! * (n-k)!)     for (int i = 0; i < k; ++i) {         res *= (n - i);         res /= (i + 1);     }      return res; }  // Function to find the nth Catalan number int numBST(int n) {        // Calculate C(2n, n)     int c = binomialCoeff(2 * n, n);        	// Return the nth Catalan number     return c / (n + 1); }  // Function to find total binary tree int numBT(int n) {        int fact = 1;      	// Calculating factorial   	for(int i = 1; i <= n; i++) {      	fact = fact * i;      }      	// Total binary tree = Tatal binary search tree * n!   	return numBST(n) * fact; }  int main() {        int n = 5;     cout << numBST(n) << endl;  	cout << numBT(n) << endl;     return 0; }

Java

// Java code of finding Number of Unique // BST and BT with N Keys class GfG {      // Function to calculate the binomial coefficient C(n, k)     static int binomialCoeff(int n, int k) {                // C(n, k) is the same as C(n, n-k)         if (k > n - k) {             k = n - k;         }          int res = 1;          // Calculate the value of n! / (k! * (n-k)!)         for (int i = 0; i < k; ++i) {             res *= (n - i);             res /= (i + 1);         }          return res;     }      // Function to find the nth Catalan number     static int numBST(int n) {                // Calculate C(2n, n)         int c = binomialCoeff(2 * n, n);          // Return the nth Catalan number         return c / (n + 1);     }      // Function to find total binary tree     static int numBT(int n) {         int fact = 1;          // Calculating factorial         for (int i = 1; i <= n; i++) {             fact = fact * i;         }          // Total binary tree = Total binary search tree * n!         return numBST(n) * fact;     }      public static void main(String[] args) {         int n = 5;         System.out.println(numBST(n));         System.out.println(numBT(n));     } }

Python

# Python code of finding Number of Unique # BST and BT with N Keys  # Function to calculate the binomial coefficient C(n, k) def binomialCoeff(n, k):        # C(n, k) is the same as C(n, n-k)     if k > n - k:         k = n - k      res = 1      # Calculate the value of n! / (k! * (n-k)!)     for i in range(k):         res *= (n - i)         res //= (i + 1)      return res  # Function to find the nth Catalan number def numBST(n):        # Calculate C(2n, n)     c = binomialCoeff(2 * n, n)      # Return the nth Catalan number     return c // (n + 1)  # Function to find total binary tree def numBT(n):     fact = 1      # Calculating factorial     for i in range(1, n + 1):         fact *= i      # Total binary tree = Total binary search tree * n!     return numBST(n) * fact  n = 5 print(numBST(n)) print(numBT(n))

// C# code of finding Number of Unique // BST and BT with N Keys using System;  class GfG {          // Function to calculate the binomial coefficient C(n, k)     static int binomialCoeff(int n, int k) {                // C(n, k) is the same as C(n, n-k)         if (k > n - k) {             k = n - k;         }          int res = 1;          // Calculate the value of n! / (k! * (n-k)!)         for (int i = 0; i < k; ++i) {             res *= (n - i);             res /= (i + 1);         }          return res;     }      // Function to find the nth Catalan number     static int numBST(int n) {          // Calculate C(2n, n)         int c = binomialCoeff(2 * n, n);          // Return the nth Catalan number         return c / (n + 1);     }      // Function to find total binary tree     static int numBT(int n) {         int fact = 1;          // Calculating factorial         for (int i = 1; i <= n; i++) {             fact = fact * i;         }          // Total binary tree = Total binary        	// search tree * n!         return numBST(n) * fact;     }      static void Main() {         int n = 5;         Console.WriteLine(numBST(n));         Console.WriteLine(numBT(n));     } }

JavaScript

// JavaScript code of finding Number of Unique // BST and BT with N Keys  // Function to calculate the binomial coefficient C(n, k) function binomialCoeff(n, k) {      // C(n, k) is the same as C(n, n-k)     if (k > n - k) {         k = n - k;     }      let res = 1;      // Calculate the value of n! / (k! * (n-k)!)     for (let i = 0; i < k; ++i) {         res *= (n - i);         res /= (i + 1);     }      return res; }  // Function to find the nth Catalan number function numBST(n) {      // Calculate C(2n, n)     let c = binomialCoeff(2 * n, n);      // Return the nth Catalan number     return c / (n + 1); }  // Function to find total binary tree function numBT(n) {     let fact = 1;      // Calculating factorial     for (let i = 1; i <= n; i++) {         fact *= i;     }      // Total binary tree = Total binary search tree * n!     return numBST(n) * fact; }  let n = 5; console.log(numBST(n)); console.log(numBT(n));

Output

42 5040

Time Complexity: O(n), where n is total number of node
Auxiliary Space: O(1)

Total number of possible Binary Search Trees and Binary Trees with n keys

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Total number of possible Binary Search Trees and Binary Trees with n keys

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