All living things exchange the gas Carbon 14 (C14) with the atmosphere around them—animals and plants exchange Carbon 14 with the atmosphere, fish and corals exchange carbon with dissolved C14 in the water.
Throughout the life of an animal or plant, the amount of C14 is perfectly balanced with that of its surroundings. The C14 in a dead organism slowly decays at a known rate: its "half life".
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It's not that there are 11000 year old trees, but there have been young trees growing happily next to old trees for that long getting the same weather-related patterns in their rings, by which we can piece together a time line.