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For archaeological finds regarding pottery and other fired archeological materials, the typical radiometric dating wouldn't be applicable; however, another method of dating known as thermoluminescence dating is.
The chances of this happening at ground level, (or if the dead plant is buried) is rare enough that 14C won't be produced in dead matter.There are a lot of options, and the specific ones used will vary from site to site.I'd just like to add that the best measurements come from using several of these dating methods.If so shouldn't my abundance of C14 be the same as an animal from 1000's of years ago? When alive, a plant is constantly taking in CO2, with a percentage of the carbon atoms being 14C.When it dies, this intake of "new" 14C stops, essentially starting the radioactive decay stop watch: If you know how much 14C was present in the atmosphere at the time, and know that no new 14C will be taken in to a dead plant, you can approximate its age to a reasonable degree of accuracy.
If radiometrically dating the rock layer the artifact was found in, a cache of bones found in the same layer, and perhaps a few other objects/samples all provide similar date ranges, we can be reasonably confident in the dating of the object(s).