On the planet of Caturn, God created $m$ male cats and $f$ female cats, each with zero stripes. At any time, two cats of opposite genders can mate, giving birth to a cat with a gender of the parents' choice.
The laws of cat genetics dictate that the number of stripes a child has will be one more than the number of stripes of its stripier parent. In other words, if the mother has $s$ stripes, and the father has $t$ stripes, the child will have $1+\max(s,t)$ stripes.
However, God issued the following two commandments:
No Incest: Two cats may not mate if they have a common ancestor.
Similarity: Two cats may only mate if their stipe numbers differ by at most one. (A cat with $9$ stripes can only mate with an $8,9$ or $10$ stripe cat.)
Assuming the cats obey these commandments, what is the largest number of stripes a cat on Caturn could possible have (as a function of $m$ and $f$)?
This is a variant of How long can a population last without incest? Other than rewording, the only change I made was adding the "Similarity" rule.