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In a comment to this question, it was said that

Radiocarbon dating won't work. For a start the tree needs to be dead, which since you're not using the rings I'm assuming it isn't. Also the atomic testing in the 20th century completely messed up the level of C14 in the atmosphere, so radiocarbon dating can't be used for anything that didn't die 70 years ago or more.

I did not know this. So, what kind of dating techniques biologists use today? :-)

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    $\begingroup$ The question involved dating a live tree. Radiocarbon dating does not work on live organisms, as they are assumed to be still in equilibrium with the environment. It measures the time after death of the organism. $\endgroup$ – March Ho Feb 22 '15 at 10:03
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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.

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