yeah, I think mud huts (or wattle and daub) had thatched roofs that hung way out past the walls to protect them from driving rain.  Of course they would need occasional touch-up from time to time.  But thatched roofs can last a surprisingly long time.  An anthropology museum near my home had info about thatched roofs....it said they will last between 25 to 100 years before they need repair!