I have seen a "total vegetation killer" in wal mart and home depot...

there is specific granular killers meant for areas like gravel drives that will kill everything for at least a year.

of course, the toxic runoff would also kill anything in its path, so you might want to consider that before you start with them.

black plastic is not the answer... you will genuinely regret it. if you're going for total weed smothering, but don't want the next tenant to curse you...courragated cardboard is a wonder cure. three overlapping layers of wet cardboard might even be enough to smother that tree (suckering trees are a pest, I know) for good.

and in three years, you'll be able to till the mulch into the soil and plant there, instead of having to rake it away from the plastic (more work than digging the bed, let me tell you)

what is the intention for this area? a playground? a seating area? a large bed that you intend to landscape once the thing is cleared of annoying volunteers?

sometimes, that can make all the difference (if you have kids or pets, poison is rarely the way to go- I'd cut the suckers off at the ankles, apply killer with an eye dropper, and then flame-weed any repeat offenders until you sapped the roots's energy reserves, and the darned thing actally DIES)