Catalan Numbers (original) (raw)

Last Updated : 18 Jul, 2025

**Catalan numbers are a sequence of natural numbers that appear in various counting problems, often related to recursive structures. They are named after the French-Belgian mathematician Eugène Charles Catalan.

These numbers have applications in combinatorial mathematics, such as counting paths, tree structures, and polygon triangulations.

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Catalan Numbers in Pascal's Triangle

The first few Catalan numbers are given here,

1, 1, 2, 5, 14, 42, 132, 429, 4862, 16796, .....,

Formula for Catalan Numbers

The n-th Catalan number **C n can be calculated using the following formula:

C_n = \frac{(2n)!}{(n+1)!n!}

Where n is a non-negative integer, and the factorial symbol "!" represents the product of all positive integers less than or equal to that number.

Alternatively, you can use the binomial coefficient:

C_n = \frac{1}{n+1} \binom{2n}{n}

Catalan Numbers Examples

Let's calculate the first few Catalan numbers:

C_0 = \frac{(2 \times 0)!}{(0+1)!0!} = \frac{1}{1} = 1
C_1 = \frac{(2 \times 1)!}{(1+1)!1!} = \frac{2}{2} = 1
C_2 = \frac{(2 \times 2)!}{(2+1)!2!} = \frac{24}{6 \times 2} = 2
C_3 = \frac{(2 \times 3)!}{(3+1)!3!} = \frac{720}{24 \times 6} = 5

So, the first few Catalan numbers are: 1, 1, 2, 5, 14, 42, ….,

**Applications of Catalan Numbers in Computer Science

Catalan numbers arise in several combinatorial problems, including:

**Parenthesization: The number of ways to correctly parenthesize **n + 1 factors (e.g., for multiplying expressions like (a⋅b)⋅(c⋅d)).
Application: Used in compilers, syntax checking, and expression evaluation.
**Binary Trees: The number of distinct binary trees with n internal nodes.
Application: In data structures like AVL trees, Red-Black trees, and splay trees.
**Triangulations: The number of ways to triangulate a polygon with **n + 2 sides.
Application: Computer graphics, mesh generation, and computational geometry.
**Paths: The number of paths along the edges of a grid that do not pass above the main diagonal.
Application: Compiler theory, token matching, context-free grammars.
**Generating Combinations: Catalan numbers are used in combinatorial generation algorithms for generating combinations, partitions, and tree structures.
Application: Used in AI, automated testing, and combinatorial optimization problems.
**Matrix Chain Multiplication Orderings: The number of ways to fully parenthesize a product of n + 1 matrices is given by the n-th Catalan number.
Application: Optimizing matrix multiplication in dynamic programming algorithms.

**Applications of Catalan Numbers

**Fibonacci Sequence

**Program for nth Catalan Number

Solved Questions on Catalan Numbers

**Question 1. How many valid parentheses, expressions can be formed with 4 pairs of parentheses?

**Solution:

The number of valid parenthesis expressions for nnn pairs of parentheses is given by the n-th Catalan number, C_n

The formula for the n-th Catalan number is:

C_n = \frac{1}{n+1} \binom{2n}{n} = \frac{(2n)!}{(n+1)!n!}

For 4 pairs of parentheses, n = 4.

C_4 = \frac{1}{4+1} \binom{8}{4} = \frac{1}{5} \times \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = \frac{1}{5} \times 70 = 14

So, there are 14 valid parenthesis expressions that can be formed with 4 pairs of parentheses.

**Question 2. How many distinct binary search trees can be formed with 3 nodes?

**Solution:

The number of distinct binary search trees (BSTs) that can be formed with n nodes is the n-th Catalan number.

For n = 3, we can use the Catalan number formula to find the solution.

C_3 = \frac{1}{3+1} \binom{6}{3} = \frac{1}{4} \times \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = \frac{1}{4} \times 20 = 5

Thus, there are 5 distinct binary search trees that can be formed with 3 nodes.

Practice Problems Based on Catalan Numbers

**Question 1. Find the 5th Catalan number?
**Question 2. How many distinct binary search trees can be formed with 4 nodes?
**Question 3. Calculate the 3rd Catalan number?
**Question 4. How many valid parentheses expressions can be formed with 2 pairs of parentheses?

**Answer:-