If so how far can the roots expand underground?
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As long as the roots are still in the ground, they will gather up the nutrients and feed the tree and eventually it will grow again, but not in the shape it used to be. You know when you see a certain variety of trees growing in random places, this could be either it seeded from another tree close by or it was previously cut down and started growing again.
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It depends upon the species of tree. Some will sprout from the stump or send up suckers from the roots such as willows. But most tree roots die once the nutrients stored in the root tissues are depleted.