What the tree growth rings gives us is less than 4000 years, which is mostly known.
Deep ocean sediment cores, lake sediment varves, coral samples, and speleothems does not give any dates to samples to verify their carbon compositions. These rely on other radioactive dating mechanisms, when the entire process is in question.
"basis for the terrestrial calibration data set beyond the beginning of the tree rings at 12.4 kyr".....
"INTCAL04 TERRESTRIAL RADIOCARBON AGE CALIBRATION, 0–26 CAL KYR BP"
It appears that they use tree rings up until 12400 years ago. I'm going to drop this topic here though, because like I said, I don't know that much about it and will not be able to hold an informed debate on dating via radioactive decay.
I would state my opinion that a lot of the older dating methods are probably less than perfectly accurate, whatever that means. That being said, if something is dated ~ 1 billion years old, it is probably not only 6000 years old. Maybe it's only 900 million.