What happens after a tree is removed?

You can leave it (and let it rot slowly), grind it below grade, or excavate it. For most yards, stump grinding is the practical choice.

A stump that keeps sending up shoots isn't dead yet. Some species, including tree-of-heaven, black locust, sweetgum, poplars and willows, sprout from the stump or roots after the trunk is cut. Cutting sprouts as they appear eventually exhausts the roots; grinding usually ends stump sprouts but not always root suckers further out.

Mushrooms over the old root zone are a sign the roots are decaying, which is what you want. They aren't a threat to the house or healthy trees. Termites and carpenter ants may use a decaying stump left near the house, which is one more reason to grind stumps close to foundations.

Large roots decay over years. As they do, the soil above can settle, especially where grindings filled the hole. Top up with soil as needed. In expansive clay regions, removing a large tree near a foundation can change soil moisture, so it's worth discussing with a foundation professional.

Choose a species suited to your region and the space at maturity, and plant away from the old stump, wires, and foundations. Many cities give or subsidize trees, and some require replacement trees as a permit condition.

Think about why the old tree failed. If it was a pest like emerald ash borer, or a species that's short-lived where you live, pick something different. Mixing species across a yard or street means one pest or storm can't take everything at once. Your state's extension service or forestry agency publishes recommended species lists by region.