I am looking for a data structure to add, remove, get, and find on categories.
For example:
A tree would be a good approach, but I get the feeling you are thinking there is going to be a one-size-fits-all data structure that you could use and that is not really how I envision it. I concur with Mark's solution, but recommend a Dictionary instead of a List. That way you can lookup a Category and get its subcategories quickly.