Before anything else: what are Catalan numbers? Well, they’re a sequence of numbers that come up all over the place in mathematics. They start with 1 (because why not), then go 2, 5, 14, 42, 132, and so on. These numbers have some pretty interesting properties for example, if you multiply two Catalan numbers together, you get a number that’s also in the sequence!

But what do they have to do with parse trees? Well, let us explain…

In computer science, we often use parse trees to represent syntax structures. For example, when you write code in Python or JavaScript, your program is actually made up of a bunch of little pieces that fit together like puzzle pieces and those pieces are represented by nodes on the tree! Each node has its own set of rules (called grammar) that tell us how it fits into the larger structure.

Now, here’s where things get interesting: when we write code, we often use recursion to break down complex problems into smaller ones. And guess what? Recursion is also used in math and algorithms! In fact, Catalan numbers are a great example of how recursive structures can be represented using parse trees.

So let’s say you want to count the number of ways you can arrange parentheses in an expression like (x + y) * z w. Well, there are actually 14 different ways to do that! And if we represent each possible arrangement as a node on our parse tree, we get something that looks like this:

________
/ \
(x + y) * z w
/ \ / \
x + y * z
| | |
–

As you can see, each node on the tree represents a different part of our expression. And if we count up all the nodes (including the root), we get 14! But wait… there’s more! If we want to find out how many ways we can arrange parentheses in an even longer expression like ((x + y) * z w) / (a + b), well, that would be a lot of counting!

Luckily for us, Catalan numbers come to the rescue. By using recursion and parse trees, we can count up all those nodes without having to do any tedious manual calculations. And best of all, it’s actually pretty easy once you get the hang of it!

We hope this has been helpful (and maybe even a little bit entertaining) for all of our fellow nerds out there. Until next time…