Hi John!
If you do not do anything else, the remaing tree is going to sprout by coppincing. The eucalyptus tree's roots are not so deep. It has no more than 2 meters down. If we wait the sproutings coming out, you may kill the tree by spraying glyphosate or other herbicide. One agronomist may help you to do that. After the tree is dead, the roots are going to decay and nothing else is going to happening to the soil underground.
Regards,
Luciano Lisbao Jr