Weeping Cherry Tree Sprouts?

I'm a 'thrifty' person by habit, scavenging for seeds whenever possible. This gives me a wide variety of flowers, bushes, and trees to enjoy the following spring. However, this year I was trying to get seeds from a weeping cherry tree and couldn't see any seeds beneath the tree. Yet, just along the driveway edge (beneath the overhang of the cherry tree boughs) were a multitude of sprouts with leaves that look a great deal like those of the weeping cherry tree.

Is it possible that these sprouts are indeed the 'offspring' of the mature weeping cherry tree? I have looked on the Internet for some kind of information to help me identify these sprouts. But nothing I've read says that the tree throws it's seed and the sprouts grow beneath it all in one season. Have I happened upon a gold mine of weeping cherry trees, or, will I be the one weeping when these things mature into some huge ordinary tree?

Hardiness Zone: 5a

Verita from Mansfield, OH