That's an interesting question. In all of the following, I am assuming computer science here, other fields may have completely different dynamics:
What do you think is the best ratio between the number of PhD students and the number of post-docs in a research group? Why?
PhD students and postdocs fill fundamentally different roles in an university. PhD students are the heart and most important part of a research group. They are where the actual science happens (everybody else is just too busy most of the time). The role of postdocs can vary between "more senior PhD student" to "assistant professor with bad job title", though in most larger research groups it tends to be more the latter than the former. As such, the most common task of postdocs seems to be to take over the day-to-day advising of PhD students from a busy professor. I would say the optimal "formula" for such postdocs is roughly (number of PhD students / 3) - number of professors. The idea being that, as long as the research group is still small, no postdocs are actually needed. In a larger group, you need roughly one postdoc to help about 3 PhD students, minus the few people that the faculty can / wants to actually supervise themselves. In reality, though, the real ratio fluctuates widely based on available funding, people finishing, postdocs moving on, etc.
Let's go over your examples:
one principal investigator professor, one researcher (~ assistant professor), zero post-doctoral fellows, and nine PhD students.
That seems a bit borderline in terms of PhD students per advisor / senior person, but probably still ok.
one principal investigator professor, one post-doctoral fellow, twelve PhD students, and no researchers.
6 PhD students per senior person seems already tough to provide good guidance, especially as the professor certainly will have lots of other things to do as well.
I recently visited a research group in a university where, surprisingly, there are one principal investigator professor, ten post-doctoral fellows and two PhD students.
I guess this is a group where the postdocs are more in the "glorified PhD student" mode. I am not a fan of this model. If the responsible person thinks that 10 postdocs are more productive than 10 PhD students, I think they are sadly mistaken. Or paraphrased from Jorge Cham "as a postdoc you can finally spend all your time on what your PhD training has prepared you to do - applying for faculty positions" (source).