How to come up the number of nodes on a given level in heaps?
You use the principle of scientific induction: you try a few examples and guess a good bound, and then try to prove it. Simultaneously, you try to construct counterexamples. Sometimes your failed attempts at proving a bound help you find a counterexample refuting it.

Often in research one of the hardest steps is to come up with the correct conjectures. This is a creative endeavor and there are no rules which guarantee success. Skill and experience help, but sometimes you just need to be lucky.