There are no serious pest problems of trees except birds and other animals that attack maturing fruit. There are some named varieties of star apple available from nurseries but they are difficult to find. The 'Haitian' variety, introduced by William Whitman, which is dark purple, is one of the more common ones seen as grafted plants. Many nurseries that propagate star apples do so simply by planting seeds. Seedlings take about six years or more to bear fruit and if you want superior varieties, plant seedlings and then try to get graft wood from mature bearing trees and do shield budding, or veneer grafting.
Trees can also be air layered but air layering usually takes four to six months to be successful.
As long as they are protected from cold winds or freezing temperatures, trees have very few problems and produce quantities of delicious fruit each year for one to enjoy.