So the carbon dating varies from sections of the tree, from inner to outer, to the bark, is that correct?
It’s called radiocarbon dendrochronology. Dendrochronology as a technique began before radiocarbon dating, but only yielded relative dates, and they were (as you might imagine) a little subjective. The technique began in the SW US where there are timbers in precolumbian structures that can have their ring spans counted (for the age) and measured (for comparison with other samples of the same species).
Radiocarbon dating used to use a lot more of the sample, so it was used sparingly; as technology improved, IOW, by about 1970, it was possible to get a more accurate date with much smaller samples, making radiocarbon dating of each ring practical. Before this, a log might be cross-sectioned, and the slice dated, yielding an average date, which works adequately because the rings could be counted beforehand.
Dating individual rings turned up the infomrmation that the RC quantity in the atmosphere varies from year to year; that in turn made dendrochronology more important, since the sequence of fluctuating values can be “wiggle-matched” to build a ring sequence beginning with living specimens, and going back to overlapping sequences in dead samples, then overlapping those with older samples, to build a continuous sequence that can be cross-checked by other labs using second samples from the same specimens.