Are the leaves about twice as long as wide, tapered at the tip; bark has high, corky ridges? 

CLUE: the fruit is purple and berry-like. 

It is a hackberry.

Are the leaves heart-shaped and about as wide as long? 

CLUE: the fruit is small and nut-like, attached to a wing-like leafy bract. 

These are basswood or linden trees.