Some hedging plants and native trees produce suckers (new plants that grow up from the parent plant's root system). Sometimes this is good - because you want a bushier plant. Rugosa roses make a bushier hedge, for example, because they sucker.
Sometimes this is bad - because the sucker is the same as the rootstock, but not the tree as the plant is grafted. An example here would be a named rose variety that is grafted onto a dog rose rootstock
Here is a list of trees and shrubs that sucker. As usual it is not exhaustive, so feel free to comment and add. But if you don't babies sprouting around your plants these are the ones to avoid. You can reduce suckering in grafted trees (and an awful lot of trees are grafted - all named apple trees for example) by not damaging their root systems. If you do see a sucker it is best to "tear" it out - cutting it back just means it will try again.
Suckering trees include
Suckering shrubs include:
Bamboos (just as a caution - we don't sell them.... yet)
Philadelphus (mock Orange)