Lets say we have some arbitrary grammar for which we would like to know how many different syntax trees does it generate. For example the following:
S -> A1|1B
A -> 10|C
B -> C1 | $ \varepsilon$
C -> 0|1
This is a quite simple language and the number of unique trees is not hard to figure out by just going through all of the different possibilities, but for more complex languages it gets difficult to count the number of unique trees by just examining it. For this reason, I’m wondering if there does exist some analytical approach, rules of thumb, or software that would allow to count the number of unique trees faster than just looking at the language long enough until all the possibilities are exhausted?