# Time complexity to find out the number of ways to parenthesize N matrices

I am trying to figure out the $$time$$ $$complexity$$ to find out the number of ways we can parenthesize $$N$$ $$matrices$$. I have approached this problem as, say if we have $$N+1$$ matrices then we can parenthesize them in $$\dfrac{2N \choose N}{N+1}$$ ways

For example, see the below image, here we have 3 matrices which we can represent in a tree and we have 2 nodes which we can arrange to form different unlabeled binary trees in $$\dfrac{2N \choose N}{N+1}$$ ways which in turn will give us the number of ways in which we can parenthesize these 3 matrices.

But the problem I am facing is that I am unable to figure out the time complexity which this method will take. Any help will be highly appreciated.

These are called Catalan numbers, and the term $$\binom{2N}{N}$$ is called a central binomial coefficient.
$$C_n = \prod_{k=2}^n \frac{n+k}{k}$$