It should be fine. Eucalyptus trees do produce allelopathic biochemicals, but those biochemicals aren't very prevalent in the bark or trunk portions of the tree and, in any case, they leach out and degrade pretty rapidly once the tree is cut down.
There is a side benefit in that bugs do not like it.
I am a strong believer in the simple fact is that what matters in this life is how we treat others. I think that's what living is all about. Not what I've done in my life but how I've treated others. ~~ Sharon Brown
