How many ways can we write a valid sequence of $2n$ parentheses? How many rooted binary trees are there on $n$ vertices? How many up-right paths from $(0,0)$ to $(n,n)$ stay on or below the line $y=x$? In this class, we will explain what these questions mean and why they are all answered by a sequence called the Catalan Numbers.

How many ways can we write a valid sequence of $$2n$$ parenthesis? How many rooted binary trees are there on $$n$$ vertices? How many up-right paths from $$(0,0)$$ to $$(n,n)$$ stay below the line $$y=x$$? In this class, we will explain what these questions mean and why they are all answered by a sequence called the Catalan Numbers.