C++ Programming – Program for nth Catalan Number
Catalan numbers are a sequence of natural numbers that occurs in many interesting counting problems like following.
1) Count the number of expressions containing n pairs of parentheses which are correctly matched. For n = 3, possible expressions are ((())), ()(()), ()()(), (())(), (()()).
2) Count the number of possible Binary Search Trees with n keys (See this)
3) Count the number of full binary trees (A rooted binary tree is full if every vertex has either two children or no children) with n+1 leaves.
The first few Catalan numbers for n = 0, 1, 2, 3, … are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, …
Recursive Solution
Catalan numbers satisfy the following recursive formula.
Following is the implementation of above recursive formula.
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Output :
1 1 2 5 14 42 132 429 1430 4862
The value of nth catalan number is exponential that makes the time complexity exponential.
[ad type=”banner”]Dynamic Programming Solution
We can observe that the above recursive implementation does a lot of repeated work (we can the same by drawing recursion tree). Since there are overlapping subproblems, we can use dynamic programming for this. Following is a Dynamic programming based implementation in C++.
Output:
1 1 2 5 14 42 132 429 1430 4862
Time Complexity: Time complexity of above implementation is O(n2)
[ad type=”banner”]Using Binomial Coefficient
We can also use the below formula to find nth catalan number in O(n) time.
Output:
1 1 2 5 14 42 132 429 1430 4862
Time Complexity: Time complexity of above implementation is O(n).
We can also use below formula to find nth catalan number in O(n) time.
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