I cut them down and spray glyphosate on the stump and that usually does the trick. Otherwise every stump that you leave has 10 suckers shooting out of it. I don't like using glyphosate around ponds if I don't have to. I use it sparingly and have never had a problem.
If you have any duck weed, “SONAR” will kill it and last a couple years.(Unbelievably Expensive, however!) I used it and it killed my “beautiful” Weeping Willows as well!!! Dealer said it wouldn’t hurt any “well-established” trees??