Cherry Question

We have a 12' choke cherry tree in what is considered 80% MY shrub garden.
(Generally my wife has taken to perennials, vegetables and expansion; I've
taken to trees, shrubs, landscaping (to keep up with my wife's expansion
plans) and water). The year after we moved in we decided the cherry had to
be tamed or removed because it was in front of the house. Five years later
it is trained to a weeping design, and has become a terrific focal point of
that grouping and the yard. The birds use it as a staging area for the
feeder and birdbath and at some point robins discover the cherries every
year and go nuts.

Because it is pruned fairly heavily, there's a tendency for it to produce
many suckers growing out of its base root system close to the main trunk.
Saplings are easy to yank out, but these suckers have to be cut pretty
frequently. Its is a maintenance problem and could ultimately be an entry
point to disease. If I wrap the base with landscape cloth and mulch heavily
will the suckers whither and die? Or will they simply push their way
through because they are attached to the main tree and keep coming?
Thoughts?

Cut Flower Tip:

Not sure where I got this, but it works really well. Before you take cut
flowers into the house, set them in water and leave them outside for about
an hour. For whatever reason, the bugs will abandon the cut flowers en
masse. This has worked wonderfully for us, and makes for a much happier
dinner centerpiece.

Bill C.