• What are Catalan numbers? Why do they occur so often in combinatorics?

• How many binary trees with n internal nodes?

  

• What are bracket sequences? How many balanced bracket sequences are there with n opening and n closing brackets?

• Mountain Ranges and Diagonal-Avoiding paths

• Recursive definition of Catalan numbers.

$$ C_n = C_{n-1}C_0+C_{n-2}C_1+\cdots+C_1C_{n-2}+C_0C_{n-1} $$

• Closed form of Catalan numbers.