I've just had to cut down a Eucalyptus tree.
Do I need to treat the stump that's left behind to stop any regrowth?
Absolutely you will need to treat it if you don't want regrowth. Most trees, provided they're healthy, will regrow from an untreated stump, but eucalypts in particular tend to respond by growing 6-8 feet in a season after such treatment. You'll need to find a stumpkiller and follow the instructions - you haven't said where you are, so it's not possible to mention particular products - they differ from country to country.