I live in America; I have 2 trees I want to cut down. My grandmother is a volunteer in a botanical garden. She said it is an apple tree and that it is sick and old, that is why it does not produce edible fruit, (it produces small things the size of blueberries), and it will never produce edible fruit. Can I cut them down?

Yes, you may cut it down. And it would have been permitted even if you would have the merit of living in the land of Israel, because the prohibition of “Thou shalt not destroy the trees” (Deuteronomy 20, 19) applies everywhere. Nevertheless in a case like yours the Rambam (Melachim 6, 9) writes that it is permitted to cut down a tree that got old and does not produce more than a small amount which is not worthwhile troubling for. The amount an olive tree should produce in order for it not to be cut down is a quarter of a Kav (approx. 0.65 lbs.) olives, and a palm tree which produces a Kav (approx. 2.6 lbs.) dates should not be cut down. However it is preferable to have a gentile cut it down if possible for fear of danger. (Chida in Chaim Sha’al A, 23).