There is a bijection between Dyck paths and rooted binary trees (see here, here and exercise 12-4 of Introduction to Algorithms by Cormen et.al. (edition 3)); they are both given by the Catalan numbers.
Now, Dyck paths tell us how to start at $0\$$ and end at $0\$$ after tossing a coin $2t$ times (when we get $1\$$ on heads and lose $1\$$ on tails) without ever exceeding $0\$$. Dyck paths can be generalized to the paths in Bertrand's ballot problem (by lifting the glass ceiling from $0$ to $k$) where we still start at $0\$$ but end at $k\$$ after $2t+k$ tosses ($t$ of which are tails) such that we never cross $k\$$. These are covered here: Probability that random walk will reach state $k$ for the first time on step $n$. I'll call these "ballot paths".
Just like there was a bijection between Dyck paths and binary trees, there should also be a corresponding bijection between ballot paths and some other tree-like structures. This is alluded to in the book Analytic combinatorics by Flajolet and Sedwick (0th edition) on page 63, section 1.5 where they refer to "forests" fitting this description.
I'm not able to place my finger on these mysterious forests and how they are constructed.