When you think of palm tree fruit you ususally think of cocnuts, but there Did you know that wax, oil, jelly and wine are also produced from their fruits. leave the dates on the trees not just to ripen but to dry some too before being harvested. Coconut is the first fruit that probably comes to mind, and date fruits might be the next. But one you may not have known was a palm fruit is the Acai berry. This fruit comes from a South American tree in the Euterpe family . seed that is the edible part of this palm, with the fruit being fairly bland and dry (I. Palm tree fruit comes from the California fan palm species native to arid desert The fruits can also be soaked, peeled off the seeds and dehydrated to dry for.