Lemons and limes grow from different trees. Lemons grow from Citrus limon trees. Key limes grow from Citrus aurantifolia trees, and Persian limes grow from Citrus latifolia trees.
Persian limes produce lots of juice and are the limes that most grocery stores sell, and key limes are sweeter and used in many desserts. There are also other lime varieties. Trees that produce both lemons and limes can be created by grafting branches from one tree onto another tree. These are called fruit cocktail trees and are possible because lemon and lime trees are in the same plant genus, Citrus. However, fruit cocktail trees are not natural and can only be created artificially.