Removing a tree is only partially accomplished by cutting it down. You must also do something about the old stump that is left behind.
Stumps shouldn't be left in the yard to rot naturally. The decaying wood can attract a termite colony, which may move on to feeding on your home once the stump is long gone. Other pests are also drawn to old stumps, from insects that can infect nearby trees to animals like skunks that may burrow out a den beneath the stump.
There are other issues with old stumps, as well. They can ruin the appearance of an otherwise well-manicured lawn, for example. Stumps may also pose a tripping hazard for visitors or playing children. Mowing and trimming weeds around a stump can be another problem that makes yard work a pain.
There are a few options for removal, such as digging out the stump, but the best option is grinding. A special mechanical grinder is used to grind down the stump to well below the soil surface. The goal is to grind out the stump so little remains other than the radiating roots. This ensures that the stump won't resprout. In species where root sprouting is an issue, the service may first chemically treat the stump to kill the roots, and then they will grind out as much of the stump as possible.
Grinding doesn't result in mulch, but it does produce a lot of sawdust. Most stump grinding services will fill in the hole with the sawdust to help level out the yard, but you can also have them haul it away if preferred.
Recovery after stump grinding takes some time. If the sawdust is left in the hole, it will need to be mixed with compost and topsoil so that it can support new plant growth. It's best to install sod or a garden bed over the old site of a tree, as a new tree may not thrive in an old stump site due to the compaction of the soil from old roots.
Be aware that the soil will sink for a few years as both the sawdust and the old tree roots decompose. You can topdress the site with topsoil once or twice a summer to help keep it built up to the desired level.
Contact a tree stump grinding service if you are ready to have the old stump removed from your lawn.