Since there are four proofs in the wikipedia article you referenced, it seems you aren't looking for a mathematical explanation for the correspondence between the Catalan numbers and the permutations of a binary tree.

So instead, here are two ways to try and intuitively visualise how the Catalan sequence (1, 2, 5, 14, 42, ...) arises in combinatorial systems.

## Dicing polygons into triangles

For a polygon of side *N*, how many ways can you draw cuts between the vertices that chop the polygon up entirely into triangles?

- Triangle (N=3): 1 (It's already a triangle)
- Square (N=4): 2 (Can slice at either diagonal)
- Pentagon (N=5): 5 (Two slicing lines emanating from a vertex. Five vertices to choose from)
- Hexagon (N=6): 14 (Try drawing it)
- ...and so on.

## Drawing a path through a grid without crossing the diagonal

In this case, the number of unique paths is the Catalan number.

2x2 grid => 2 paths

```
_| |
_| __|
```

3x3 grid => 5 paths

```
_| | _| | |
_| _ _| | _| |
_| _| _ _| _ _| _ _ _|
```

4x4 grid => 14 paths

5x5 grid => 42 paths

and so on.

If you try drawing the possible binary trees for a given N, you will see that the way the tree permutes is just the same.

Your desire not to just blindly accept the correspondence between the tree and the sequence is admirable. Unfortunately, it's difficult to go much further with this discussion (and explain why the Catalan formula 'happens to be' the way it is) without invoking binomial mathematics. The Wikipedia discussion of binomial coefficients is a good starting point if you want to understand combinatorics (which includes permutation counting) itself in more depth.

Very interesting question, though I'm not sure it's really programming-related :-/ Seems like more of an abstract math (topology) thing. – David Z – 2009-08-30T01:47:40.960

Uh, this has nothing to do with topology! – ShreevatsaR – 2009-08-30T02:14:02.520

@Sergio: What is your question? Are you wondering why the number of binary search trees with N keys is the same as any of the quantities shown on that page to be Catalan numbers, or are you wondering about the expression for Catalan numbers themselves (which is proved on the page in four ways)? – ShreevatsaR – 2009-08-30T02:15:14.297

1I'm wondering how to determine the number of possible BSTs that can be created given a number N of nodes. I discovered that the answer to that question is the same as asking "What's the Nth number in the catalan sequence?", but even though the formula is readily available on that wiki article, I'd like an explanation of why it works. I don't like to accept methods without some explanation of their inner logic or some basic, intuitive description other than a formula. – Sergio Morales – 2009-08-30T06:42:01.883

This applies to binary trees in general, it has nothing to do with the specifics of search trees. – starblue – 2009-08-30T10:45:29.443

1@starblue: you're quite wrong – Eli Bendersky – 2010-05-18T05:05:49.667

Just FYI - Related link: http://codingworkout.blogspot.com/2014/08/all-possible-paranthesis.html

– Dreamer – 2014-08-05T21:13:45.130