Answer: There is a product called "Sucker Stopper" which is applied to the area where suckers are formed early in the spring. According the the manufacturer it prevents suckering. I'm not endorsing the product because I know nothing about it other than what I've read. I'm not sure where you can purchase it.
You may have to dig down closer to the tree and remove the roots that are producing suckers. This will only stop the suckering for a little while, though, because it is the nature of the cottonwood to colonize an area by sending out roots and sprouting suckers.
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