Soursop trees come from tropical regions and enjoy the climatic conditions that are present there. As such, soursop trees love to have plenty of humidity in the air and are partial to warm weather as well. Soursop trees grow in hardiness zones 9 through 11 and cannot survive when the air temperature drops to near 30 degrees Fahrenheit. Based on those preferences, people living in southern states have the best chance of growing a healthy soursop tree outdoors.