How can I remove duplicate files across separate directory trees?

fdupes - finds duplicate files in a given set of directories

fdupes new/ old/

-d for delete

-N for don't ask

fdupes -N -d new/ old/

Of course! Check out FSlint:

FSlint is a utility to find and clean various forms of lint on a filesystem. I.E. unwanted or problematic cruft in your files or file names. For example, one form of lint it finds is duplicate files. It has both GUI and command line modes.

FSlint is free and open source software.