Just noticed this, but intriguingly, Catalan numbers are (2n C n)/(n+1), which hints at a connection with trees.
Off the cuff, notice that the diagonal has n+1 intersection points, and a path that never passes through the diagonal gives a forest via the isomorphism with ballot sequences [0]. Any sequence that does pass below the diagonal can be "rotated" into one that doesn't, and so there are probably n+1 paths in each "path class" on average.
Conversely, this would suggest that all paths contained in just one upper or lower triangle of the square can be counted by the Catalan numbers. Indeed, a 2x2 square has just 2 such paths and (2n C n)/(n+1) = 6/3 = 2.
[0]:https://blog.wilsonb.com/posts/2026-02-27-easy-random-trees....
Funny, but I've recently started a blog to keep my math and physics muscles working and this exact problem is the first post I've written - it is useful in high order perturbation theory in quantum mechanics that I'm planning to describe down the line. Counting these paths is cool problem but generating all of them was more challenging.
Anyway, here is the post https://kpatucha.github.io/posts/Dyck-paths-Raneys-lemma/