Your spindly trees are young mulberry trees M. rubra (red mulberry or American mulberry). Mulberry trees produce edible berries.

Mulberries are fast-growing when young, but soon become slow-growing and rarely exceed 30–50 ft. tall. The leaves are alternately arranged, simple and often lobed and serrated on the margin. Lobes are more common on juvenile shoots than on mature trees. The mulberry fruit forms in clusters, approximately 3⁄4–1 1⁄4 in. long. Immature fruits are white, green, or pale yellow. In most species the fruits turn pink and then red while ripening, then dark purple or black, and have a sweet flavor when fully ripe. The fruits of the white-fruited cultivar are white when ripe; the fruit of this cultivar is also sweet, but has a very bland flavor compared with darker varieties.

Birds relish the fruit and readily spread the seeds. Many people dislike mulberry trees because of their aggressive, “weedy” nature and high maintenance
requirements (pruning, mess cleanup, etc.). Mulberry leaves take a few different (but similar) shapes.