To be completely truthful, the growth of tree roots completely varies by season depending on each on individual species. However, most of the trees here in North Carolina experience a growth spurt for their root systems during the spring and autumn Times the year.
We know this from plenty of research conducted by arborists that have studied root growth throughout the USA within forests. For root growth to occur there needs to be plenty of watering, fertilization and generally a good amount of nutrients. In areas where droughts are a large problem, you’ll actually see things like exposed roots and further damage. This is a complete result of the lack of nutrients that a trees root system needs to absorb in order to grow and remain healthy.