Ubuntu :: Correct Option To Merge Directory Trees?

I have a couple directories in the following format:
+ Dir 1
----+ Dir A
-------- file1
----+ Dir B
-------- file2
+ Dir 2
----+ Dir A
-------- file3
----+ Dir C
-------- file2
+ Dir 3
----+ Dir B
-------- file2
----+ Dir C
-------- file1

My desired structure is:
+ Dir Final
----+ Dir A
-------- file1
-------- file3
----+ Dir B
-------- file2-1 [or something so that the two files don't overwrite]
-------- file2-2
----+ Dir C
-------- file2
-------- file1

Basically, I would like to combine all the directories that I can, without deleting/overwriting any files. I have looked at Rsync, but I could not find the correct options to do what I wanted.