ANSWER: Palm tree is short for palmetto tree. SECOND ANSWER: Not quite right. Palmetto means"little palm". As I understand it, palm means hand-shaped (think of the palm of your hand) because the leaves of a palm fan out like the fingers of a hand.
"The palm" may refer to a part of the body or to a tree. The palm tree is a type of evergreen tree which grows in tropical regions. The palm of the hand is the are of that hand that the fingers touch when one makes a fist. It is the inside area of the hand.