Catalan Number Calculator
Calculate Catalan numbers and their combinatorial interpretations including binary trees and lattice paths.
Inputs
Results
Catalan Number C(n)
42
Full Binary Trees (n+1 leaves)42
Monotonic Lattice Paths42
How to Use This Calculator
- Enter n to compute the nth Catalan number.
- Review Catalan Number C(n) — the count of valid parenthesization sequences and monotonic paths.
- Check Full Binary Trees (n+1 leaves) to see the structural interpretation.
- Use Monotonic Lattice Paths for counting grid-path problems in combinatorics.
- For large n, note that Catalan numbers grow as 4^n divided by n^1.5 times sqrt(pi).
How the result changes with n
| n | Catalan Number C(n) |
|---|---|
| 3 | 5 |
| 11 | 58,786 |
| 20 | 6,564,120,420 |
| 27 | 69,533,550,916,004 |
What each input means
- n
- Index of the Catalan number to compute (C_n). Maximum 30 to avoid overflow.
How this is calculated
Worked example, using the default values
- Identify Input Parametersn = 5 = 1 input(s) provided
- Calculate Catalan Number CCatalan Number C42 = 42
- Calculate Full Binary TreesFull Binary Trees42 = 42
- Calculate Monotonic Lattice PathsMonotonic Lattice Paths42 = 42
Engine last updated . Checked against 5 independently-derived tests — how we verify calculators.
Related Calculators
The questions that sit next to this one — chosen by subject, including calculators filed under a different category.
Discrete Mathematics
Permutations & Combinations Calculator
Calculate permutations and combinations with or without repetition for any n and r values.
Discrete MathematicsBinomial Coefficient Calculator
Calculate nCr and nPr values with Pascal's triangle row visualization.
Discrete MathematicsChromatic Number Estimator
Estimate the chromatic number of a graph using Brook's theorem and greedy coloring bounds.
More in Math & Statistics.