This kinda requires a multi part answer.
Generally speaking, for most hardwoods, to hang my rope from the branch, the branch must be at least as big as my arm where it connects at the shoulder. And even then, I'll only go out from the trunk about 2'. Tops.
Of course, the bigger the diameter of the branch, the farther out from the trunk I'll hang my rope,...IF I KNOW THE TREE. Can't stress that part enough.
The 1st time I climb a tree, I always run my rope around the trunk, so I can inspect it on the way up. Only after I'm really confident in the soundness of a particular trees structure will I "go out on a limb". Sorry, I couldn't help myself on that one.
Now, If my rope is going around the trunk, I'll allow the limb that's holding my rope in place on the trunk to be about as small as my thumb. NOTE...I was a working arborist for a number of years, and I know from a lot of experience that this works for me. I'm not telling you to do this. I'm just sayin. You do with that info what you will. YOU ARE RESPONSIBLE FOR YOU.
For the record, I go around 190#'s