math - The possible number of binary search trees that can be created with N keys is given by the Nth catalan number. Why?

Question

This has been bothering me for a while. I know that given N keys to arrange in the form of a binary search tree, the possible number of trees that can be created correspond to the Nth number from the Catalan sequence.

I have been trying to determine why this is; unable to find anything that might even attempt to explain it intuitively I resort to the collective knowledge of SO. I found other ways to calculate the number of possible trees, but they seemed less intuitive and no explanation was offered beyond how to use it. Plus the wiki page (that link above) even shows an image of the possible tree formations with 3 keys, which would lead me to think there's a nice and neat explanation to be heard (which is, needless to say, not included in the article).

Thanks in advance!

Answer 1

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.