House rules or suggestions feature on this page.
For those of you without the rules, a word of explanation. The Root Maximum (p9 Explorer Book) is a game mechanic to try and simulate that a beginning character will have a smattering in a science or skill which can be used with any subdivisions of that skill until they become more experienced in that field. As the PC gains experience in the skill (s)he will rise in the skill until they reach the Root Maximum, and from then on they will need to open up separate subdivisioins of the skill. The root maxium for a skill will depend on which category it falls under (Communication, Agility, Perception, Knowledge and Technical) which is abstracted as the sum of two of the PC's characteristics that govern that category: Agility = STR+DEX; Communication = INT+APP; Perception = POW+CON; Knowledge = INT+EDU; and Technical = DEX+INT.
For example, lets take physics for a PC named Cornelius Wang with INT= 16 and EDU= 25. He will have a root maxium equal to his INT+EDU of 41. This means that during character generation, the player can put up to 41 points in Physics without having to split the skill into it's subdivisions (which are as numerous as you can make them, such as X-Ray Crystallography, nuclear physics, biophysics, optics, etc,..). If the player wants the PC to go above the Root Max in Physics then (s)he will have to split the skill into subdivisions, with each subdivision of physics taking development points which will not increase any of the other subdivisions of Physics.
The root maximum rule is a bit clunky and slows down character generation in Ringworld as the player is confronted with a huge number of possible subdisiplines to choose from. It also makes it difficult to have a character that has an appropriate skill to solve a problem and Ringworld relies on investigation and problem solving to some extent. I propose that the root maximum instead become more like the skills bonus in Runequest and Stormbringer and to scrap the subdivisions of the major sciences. There is only a need for Biology, Physics, Chemistry and Maths. Botany and zoology would fall under Biology. Psychology is a separate discipline, though one could say that a biologist has an insight into animal behaviour.
I graduated with a degree in Biophysics and I wouldn't think of Biphysics being a subdivision of Physics. It's actually a interdisciplinary science which is difficult to render in an rpg setting. The skills and knowledge I acquired as a Biohysicist spread across Biology, Physics, Chemistry and Mathematics and included Anatomy, Neurophysiology, X-ray crystallography, Molecular Biology, Biochemistry, to quote just a few subdisciplines of the major sciences. So you see the subdivisions of the main sciences don't apply here. If anything, a biophysicist in Ringworld would have skills in Biology, Physics, Chemistry and Maths. If a player character has Physics then that character will have knowledge of all subdisciplines of Physics (rather like the rules in Call of Cthulhu) and would know a bit about X-ray crystallography. A PC with Biology would know a bit of genetics and biochemistry. This becomes easier to GM and also means that the PC is more likely to have a relevant skill to help out in an situation.