Hi there, yes, it's definitely Eucalyptus. I have three, one of which is a bit temperamental - it needs constant watering when it gets hot! The others are almost maintenance free. Your mother-in-law's tree looks as if it would benefit from an organic feed such as blood fish and bone, plus a drenching of the leaves with seaweed extract, and lots of water! There's some very good plant care information on this site: Buy Eucalyptus Tree UK | Hardy Eucalyptus Grafton Nursery (hardy-eucalyptus.com) Hope this helps.
Its a bit late for this year, but next March I would prune it down quite hard. Then give it a feed and water it well and you will get lots of nice juvenile foliage.
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Buy Eucalyptus Tree UK | Hardy Eucalyptus Grafton Nursery (hardy-eucalyptus.com)
Hope this helps.