As a wonderful and helpful begonia grower who used to post on this site used to say--'all begonias have fibrous roots' which I assume is what you mean. I guess the one found most often that would do well on your balcony are the 'Dragon Wing' types--but many of the canes and shrub-like ones would do well also. There are lots and lots of begonias, and many are easy and rewarding. It would make a difference what the sun is like too--if it's a north-facing balcony, for instance, you'd do better with some of the rhizomatous ones, which prefer more shade. If it's south-facing--well, maybe back by the building-- Really need more information!