Also, don't think that the tree is killed, rather that the stump survives and grows a new 'top'... though I may be wrong.
Nope, after the actual tree is cut down, if the tree farmer does not uproot the tree to plant new ones, the stump will rot away. The situation at my mother's house, where the tree farmers decided not to replant in a particular area, are the undesirable trees (not full, growing as two conjoined trees, etc.) remaining and eventually producing the next generation of trees.
Don't blame them for not re-planting, difficult to get to. However the view is beautiful.