If you chop it down enough times, that will kill it without needing any chemicals. You can also girdle it (remove the bark deeply in a complete circle) which will also kill it, and then you could remove it.

Boy, sometimes that can be tough. I was going to suggest girdling also, but I can also attest to the fact that some trees/ shrubs will go ahead and "sucker off", meaning they will send up additional shoots off the root system if you cut the original growth out. I had the WORST time with a single russian olive here in my own woods. I cut it off, sprayed the stump, then built a fire over the stump, then tried to pull out the remains and it STILL puts out small shoots. Granted, every year there are less, but you still have to be diligent and get those stragglers too.
Eventually you will get it done.

Some Indian tribes would girdle a tree then yell at it each day. The belief was the abuse would cause the spirit of the tree to leave and it would die. Since it's annoying you, it should be easy to think of things to shout. Of course what your neighbors will think... :)

DG, I can attest to black cherries being fairly good at suckering even after you chop the entire tree down. the stump will just sucker away as if nothing happened. Good thing though as they are good wildlife trees.

try this (which works for me):
cut the unwanted sapling to 6-8" or whatever height you can manage and still reach the top of the stump. using a throw-away brush, IMMEDIATELY paint the top of the stump with a brush killer (ortho brush-b-gone is okay). cover the stump with aluminum foil to keep rain out.

the herbicide will kill the stump, but because you are applying it only to the sapling, it won't touch the ground, infiltrate the soiland be taken up by desirable plants or affect your soil fauna. no runoff, either. this does really, really work.

it's important to treat right away to get the best absorption. even five minutes later, the cells can seal themselves and you won't get good results. good luck.

I did something similar to elke except with a hand drill, I put several small holes in the stump as deep as the drill bit would go. Poured a product - Stump Out, Stump Gone, something like that - down into the holes. I was hoping they would be absorbed by the root system since it was a tree that was terrible about sending out new shoots even yards away! It worked!! And the plants even a few inches from the trunk weren't harmed.

If you use a stump killer you could tie a piece of black plastic over the top to prevent creatures or children from coming in contact with the chemical.

chiroptera_mama, you are so funny! You make me laugh! I want to do that to a couple of trees around here but they are close to the boarder of my and my neighbors property. lol!

-Annette