Trash Trees

When I bought my house it had old growth forest in the back (cicada heaven) one of the trees is a 100 foot tree of heaven.

It spreads babies every where I am constantly removing them - had trouble with a tree that grew up in between a wall and a fence. So I pulled the fence out and have been eradicating the trees there but they keep resprouting. I have roses planted and I don't want use chemical weed killers - I have to kill the weed trees before I can put the trellis up. Also I don't use chemicals in my garden.

Has anyone had an luck with putting a trash bag over a stump?
Does anyone have another idea?

I dopubt very much that a bag over the stump will do anything, the suckers will just carry on growing regardless.

Here's a link that might help regarding completely organic removal of tree stumps, unfortunately, which ever way you look at it,, it's going to be a lot of work! http://www.hdra.org.uk/factsheets/gg11.htm

Step one- have the alianthus tree removed! It is NOT a native and is spreading all over (at least in the mid-Atlantic) and winning the competition with our natives. It has little, if any wildlife value that I know of.
Step two- invite a few strong friends over for a digging party- and spring for a hearty dinner.