You may remember the song about picking up paw paws and putting them in your pocket, though I would not recommend it as they get very squishy. The fruit referred to in the song is from the paw paw tree, and its botanical name is Asimina triloba.
It is a member of the custard apple family, made up primarily of tropical fruit trees such as the custard apple and soursop. The paw paw is the most cold-hardy species in its family. It is native to woodlands from Michigan to New York state and south to Florida and Texas and is hardy through zone 5.
There are other native paw paws, A.augustifolia and A.obovata, but they are only found in zones 8 and 9.Their flowers are more showy than the flowers of the cold-hardy A. triloba, which are a reddish /purple color with three large petals and three smaller ones.
Paw paw trees have large leaves that hang down and turn a clear yellow in autumn. The fruits are oval in shape, and they are quite soft with light yellow flesh that is sweet and has a creamy texture. They break open easily but their light brown seeds are, like the custard apple's black seeds, not edible. Paw paw trees produce suckers but these can be removed to create a single, rather than multi-stemmed, tree. The flowers must be cross pollinated to bear fruit well.
Reference: The New Encyclopedia of American Trees edited by Tony Russell.
Note: In the state of Queensland, Australia, where I grew up, the fruit of the Papaya tree are also called paw paws.