You might use it to evaluate the performance of programming languages:
since recursive methods are pretty useful for benchmarking. In particular, deep recursion tests the speed with which a language can make method calls. This is important because modern applications have a tendency to spend much of their time calling various API functions.
You will also need it to solve ballot problems like this one:
Suppose A and B are candidates for office and there are 2n voters, n
voting for A and n for B. In how many ways can the ballots be counted
so that B is never ahead of A?
or just to generate Takeuchi Numbers:
0, 1, 4, 14, 53, 223, 1034, 5221, 28437, ...