The Ponytail Palm Tree Oddly enough, a ponytail palm tree is neither a palm nor a tree. In fact, it is a member of the Agave family and is actually a succulent. Other common names for this plant include the bottle palm tree or the elephant foot tree. In the past, it has been classified as either Nolina recurvata or Beaucarnea recurvata, but the latter is now the correct classification of this plant.