Does anyone know anything about gardening?

If you do get rid you might want to check that the roots haven't already got into the foundations.
We pulled out of a house sale some years ago because a Eucalyptus tree had been removed and the seller couldn't produce the paperwork to prove that the under pinning had been done.
The house we did move to also had a massive Eucalyptus on the neighbours side of the fence but it was a good distance from the house.The roots seemed to spread far and shallow and it shed huge sheets of bark every year . Someone asked if they could have the bark to burn.