And are you truly willing to delete all backup files whose originals may have been deleted by some other user (of course, me and thee would never inadvertently delete something vital just before a backup, but other users? )? Murphy's law obtains. oops, I guess that even applies to me and thee.

Your description and the uses of rsync mentioned by others do what you asked, but unless you have some master backup, or grandparent backup intact, you're begging for grief.