The branches were cut off, the trunk was cut down, the stump grinding has finished, and now the tree is no more. It might have been pulverised into wood chips on your property, or removed in pieces, but the tree is no longer there. Or is it? Just because you can't see it, that doesn't mean that there isn't a significant section of the tree still there, just beneath the surface. Removing a tree on your property doesn't usually involve removing its root system too, unless that root system is particularly shallow. For all other trees, you might be wondering—is it important to remove the roots as well?
The Angle of the Ground
There can be some benefit to leaving the root structure of the tree buried where it is, but this depends on the angle of the ground. If the tree was growing on a slope, its root structure will help to stabilise the soil, which can prevent erosion in the event of heavy rainfall. This is unlikely to be a pressing concern if the tree was removed from a more-or-less level surface. But still, it poses the question—do you need to do anything about the roots?
Whether or not any further action is required will depend on the species of tree that was removed. The tree is unlikely to have an overly aggressive root system (since the list is fairly short, but more about that in a moment), meaning that the buried roots, when deprived of their ability to receive nutrients via sunlight (since the above-ground portion of the tree has been removed) will begin to decompose, enriching the surrounding soil as they do so.
A Second Shot at Life
Trees with an aggressive root system will essentially fight to be reborn. Poplars and willows are amongst the worst culprits, but it's important that your tree removal company identifies the tree before it's removed so they are able to tell you whether the roots will keep sprouting after the trunk has been removed. This can be problematic, since the sprouts can emerge from the ground from any section of the roots, and not necessarily where the main trunk was located. These sprouts can be eliminated on a case-by-case basis, or the entire root system can be treated with herbicides, after which it will begin to decompose.
In most instances, the root system of a removed tree can be left where it is, but you can't entirely discount the possibility that the tree will make another attempt to grow. Contact a company that offers stump grinding services to find out if they can also help you remove the roots of your trees.