If you can modify the classes, make each class implement a
»deepCopy« method instead (This should also accept a Map of
objects already copied, if the data structure should contain
cycles).
Why use a Map instead of a Set for this?
I do prefer this to using the clone() method, as you'll be reminded
when writing recursive deepCopy() methods that there's no deepCopy()
on List, etc.; writing a recursive clone() method you may well
unthinkingly invoke clone() on a collection and get a shallow copy
instead of a won't compile as a result. Obvious bugs are better than
subtle bugs, especially ones that may initially be completely silent.
You also don't need to copy Strings and other value objects that are
not mutated, unless there's a need for the original and the copy of
the value object to compare != for some reason. Otherwise just copy
the reference.
Mutable objects that are semantically-constant might also not need
copying. (E.g. a protocol handler object that is stateful, but where
this is logically always the same object and it makes sense for both
copies of your larger structure to share the same one rather than use
two separate ones.)