Catalan Numbers Explained Visually: From Recurrence to Closed Form Intuition

Catalan numbers are one of the most beautiful and recurring sequences in mathematics. From valid parenthesis strings to binary trees, to stack-sortable permutations, this single sequence appears in countless combinatorial problems.

In this video, I walk you through:

🔹Why Catalan numbers are special to me personally
🔹The recurrence relation and how to visualize it through tree construction
🔹An intuitive way to derive the closed-form formula using valid parentheses strings
🔹Why not all balanced strings are valid, and how we use a clever bijection to subtract invalid ones
🔹And finally, how all these diverse problems are connected through the same magical sequence

✨ You’ll walk away not only knowing the formula, but also understanding it deeply.

⏰ TIMESTAMPS:
00:00 - Why Catalan Numbers are special to me?
01:16 - Some Catalan number problems
03:30 - Catalan number
04:25 - Recurrence relation
05:23 - Binary tree construction using recurrence
06:56 - Closed Form Solution
07:30 - Relating 2 Catalan number problems
10:49 - Total number of parentheses strings
12:40 - Counting invalid strings
17:36 - Total - Invalid = Catalan number


📎 For those interested in deriving the closed-form using generating functions, check out this detailed explanation:
https://math.stackexchange.com/questi...


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