The field of radiocarbon dating has become a technical one far removed from the naive simplicity which characterized its initial introduction by Libby in the late 1940's.
It is, therefore, not surprising that many misconceptions about what radiocarbon can or cannot do and what it has or has not shown are prevalent among creationists and evolutionists - lay people as well as scientists not directly involved in this field.
The second characteristic of the measurement of radiocarbon is that it is easy to contaminate a sample which contains very little radiocarbon with enough radiocarbon from the research environment to give it an apparent radiocarbon age which is much less than its actual radiocarbon age.
First, any instrument which is built to measure radiocarbon has a limit beyond which it cannot separate the signal due to radiocarbon in the sample from the signal due to background processes within the measuring apparatus.
Even a hypothetical sample containing absolutely no radiocarbon will register counts in a radiocarbon counter because of background signals within the counter.
Long tree-ring chronologies are rare (there are only two that I am aware of which are of sufficient length to be of interest to radiocarbon) and difficult to construct.
They have been slowly built up by matching ring patterns between trees of different ages, both living and dead, from a given locality.