This is very typical of all plants in the genus 'Prunus'. In fact, the heavier you prune the upper branches, the more suckers you'll see. Keep clipping the suckers as low to the ground (or close to the trunk)as possible and avoid hard pruning of the rest of the tree.

Thanks for the info, Transplanted. I guess I will keep on doing what I am doing and no more pruning. It is a beautiful sight in Jan and FEb. Nancy