I have a eucalyptus tree that is approx 10 years old that's not looking too good. The leaves have turned brown and are dropping off, but there is approx 6 new shoots at the bottom of the tree that has strong new growth. Am I best letting the new growth grow and cutting off the top of the tree or should I cut these off hoping the rest of the tree would recover? Hope you can help.
If any part of the tree is alive, then you can prune off the dead growth and new growth should come back. This article will help determine if your tree is still alive: https://www.gardeningknowhow.com/plant-problems/environmental/how-to-tell-if-a-plant-is-dead-and-how-to-recover-an-almost-dead-plant.htm
There are several things that could cause this in your tree. If you have had colder than usual winters for the past 2 years, as much of the country has, the stress from the cold may have hurt the tree. It could also have a pest or a fungus. I would recommend treating the remaining tree growth with neem oil. It will kill both pests and fungus on the tree. You may also want a local tree expert to come out and look at the tree to more accurately diagnosis the problem.