Number of different binary search trees storing n distinct keys?

Question

How many different binary search trees are possible that store the values 1,2,...,n ?

So far I found a recursive formula for the number (by case distinction what's at the root):

$ T(n) = 2T(n-1) + \sum_{i=2}^{n-1}T(i-1)T(n-i), n > 1 $ and $ T(1) = 1 $

But I have no idea how to solve this recursion. Our task was only to find the recursion and I believe this to be a correct solution. But I am very interested in a closed formula of it. Can anyone link me to some resources/books or give a general hint on how it can be solved?

Answer 1

The solution to your recurrence is $$ T(n) = \frac{(2n)!}{n!(n+1)!}, $$ also known as the Catalan numbers. The quickest way to find this is by computing a few elements of the sequence and using the OEIS to identify the sequence.

Answer 2

Any time you see something resembling a convolution, that suggests generating functions as a method. Convolutions are things that look like $\sum_x f(x)f(\overline{x})$, where $\overline{x}$ denotes some kind of "complement" or "opposite" of $x$. Generating functions often turn convolutions into products.

Possible references are:

• Herbert S. Wilf, generatingfunctionology. 3rd edition, A. K. Peters, 2004. (A PDF of the second edition is available from the author's website.)

• Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics. Cambridge University Press, 2009. (PDF on the first author's website.)

• Our own reference question.