Enough would survive to drive you nuts, and once they get a foot tall or so they will be much harder to remove than when they are smaller.

I have never had a problem keeping the zillions (I think because I've never counted them) of Maple tree seedlings that sprout in the lawn every year but they are easily mowed and then because they have no leaves to use to make nutrients they die. In the well mulched planting beds simply stirring the mulch is enough to disrupt the growth cycle and those seedlings die.

What kind of tree?