About modern dating techniques
In order to date the age of a tree, you have three solutions :

If you can cut the tree then you can simply count the number of rings. Note that for very old trees the central rings will have rotted and you will only have a lower bound on its age.

If some parts of the tree are dead, you can use radiocarbon dating to estimate the time of death with a relatively bad precision (the error is about 100 years which can be a lot for a tree)

You can estimate its growth rate (either using the width of the rings, or by doing multiple measurements for several years), and deduce the age from its size.