Hmmh, interesting. Like the American relative, Chinese elm is prone to diseases. As a result, I donÂt see many at all. The image below is a rare 10 year old Chinese elm in Beijing, China. If you look at the grass nearby, there is no sucker.
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I think there maybe two possibilities:
(1) You have another elm; or you have a unique strain.
(2) Some chemical (such as miracle grow for the lawn) in your yard caused the abnormal behavior.

Are you certain it is Chinese (Ulmus parvifolia) and not Siberian elm (Ulmus pumila)? Siberian self seeds very readily and can be invasive. To get rid of the ones you don't want, cut them down close to the ground and just mow the lawn areas regularly, grind out the stumps of the larger ones. Hand pull seedlings in garden areas.

Casia,

I agree. Siberian elm is sometimes erroneously named as Chinese elm. Siberian elm is tall and upright. If you have a mature tree, the difference is obvious. The Chinese elm is naturally a weeping tree. It looks like a willow. The tree is not tall but creating lots of shade.

Of considerable horticultural merit, it was described by Hilliers as "one of the most splendid elms, having the poise of a graceful Nothofagus". The Chinese elm was introduced to Europe at the end of the 18th century as an ornamental, and is found in many botanical gardens and arboreta.

In the United States, it appeared in the middle of the 19th century, when many American elms died. The Chinese elm has proved to be resistant to Dutch elm disease. Established Chinese elm tolerates both draught and pests.

Image below - a 100 year-old Chinese elm
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Snasxs and Casia: Thank you for your reponses and also for reminding me to put more faith in my research. My next door neighbor (she's been there 21 years) told me they were Chinese Elms, after some research I suspected Siberian Elms. She was most insistent and since I was unfamiliar with Elms I conceded. Anyway.... No Miracle Grow in the lawn and no seedlings. These all seem to come from the roots of the parent. I've pulled (they don't come up) and I've dug only to discover they are growing out of a much larger root coming from the direction of the parent. I inadvertent got brush killer overspray on one and a week or so later the parent got some leaf die off. I swear if the parent wasn't so healthy and beautiful (as much as an siberian elm can be) I would cut it down so I wouldn't have to deal with all these other trees.

Alexis,

I have to correct myself. The Chinese elm can grow in up-right form. Nevertheless, it is easy to differentiate Chinese elm from Siberian elm. If you really have a mature Chinese elm, the most significant merit is unmistakable. There is a reason why Hilliers considers them of significant horticultural merit. The image below is a close-up of the trunk of a Chinese elm:
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Snasxs: No, I definitely have a Siberian Elm. I wish it was the Chinese Elm. That picture is lovely. Thank you so much for helping me with this. I just wish mine would quit sucking from the roots.

It is not your neighborÂs fault. I know some vendors intentionally mislabel the elms because the price of Chinese elms is higher. Always ask the seller: "Is the tree also called lacebark elm?"

A large tree in Texas
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snasxs, such a lovely tree. thank you

Alexis,

According to Ohio State University, Bulletin 700-00: the Chinese/Lacebark elm is tolerant of Dutch elm disease, elm yellows, and elm-leaf beetle.

The true Chinese elm (U. parvifolia) develops into a better tree for ornamental plantings than the Siberian elm. Care must be taken to ensure that cold-hardy seed sources are used. ÂOhio,' ÂDynasty,' and ÂPathfinder' are clones that have proven cold-tolerance. This tree is an excellent ornamental for Ohio gardens because of its glossy foliage, fall color, attractive bark, resistance to elm-leaf beetle, and disease-tolerance.