Eucalyptus trees

I have a Eucalyptus in my garden, 50 feet from the house, however it has grown to over 30feet high.  It has quite a few branches that seem dead, with small new branches growing from the trunk, I do not know, should I cut all the dead branches off.  When I was in Australia I did see many trees the same, they all looked very sparse.  My tree is the tourist tree the bark goes very red and peels off, I don't know the real name.  It has taken 20m years to  its current size.  I do not want to loose it.  What do I do. ???