What is the study of tree rings called?

Dendrochronology, and it does two jobs at once. A ring is a year, so counting them dates the wood; and the pattern of thick and thin rings records the growing conditions, so the sequence acts as a fingerprint that can be matched between overlapping samples - living trees to old beams to archaeological timbers - building continuous chronologies going back over 12,000 years in some regions.

That has made it the backbone of dating for other methods. Radiocarbon dating produces dates that drift, because the amount of carbon-14 in the atmosphere has varied, and the correction curve used to convert a raw radiocarbon result into a calendar date is built from tree rings of known age. Almost every archaeological date you read has been calibrated against wood.

It also catches things nothing else does. The 993 cosmic ray event left a carbon-14 spike in tree rings worldwide, which is how the Norse settlement in Newfoundland was dated to exactly 1021. Rings record volcanic winters, droughts that coincide with the collapse of civilisations, and in one case demonstrated that a Stradivarius could not be genuine because the wood post-dated his death. The oldest known individual tree, a bristlecone pine, has over 4,800 rings.