Basal shoots of trees, such as the one your pictured with the leaves, are often somewhat poor tools for identification. The leaves in these shoots can deviate from the actual leaf shape, size or color of the main branches in a tree. This depends on the species, though, but for identification purposes you always need to use the leaves of the higher branches and not the basal shoots.
Those fruits are also very easy and important tools for identification. Just saying that the are 3/8" big is pretty useless info. They could be unripe and grow up to 10"... Take a picture of the fruits, if possible. They may be laying on the ground if the tree is too big to reach up. Mulberries are very easy to identify by the fruits.