To tell a red mulberry tree from a white mulberry tree, look to the leaves. When ripe, the "edible but messy fruits" of the white mulberry — a naturalized species from Asia that has largely supplanted the native red mulberry, writes Edward Sibley Barnard in New York City Trees — can, in fact, be "white, pink, red, or purple." But while red mulberry leaves are rough on top and hairy below, notes Sibley, white mulberry leaves are smooth on both surfaces, occasionally with hairy veins.
Consensus has it that the fruits of the red mulberry have a fuller flavor, when they can be had. No luck this day, as you can see.