The growth of trees record the level of rainfall they received. There wouldn't be a reason for the tree to continue growing without showing evidence. when the tree dies is when it stops growing, not before.

We can date wood from lumber, even when it's from an archaeological site hundreds of years old. This is because we actually have a databank or library of tree rings. We can compare the exact measurements of each ring to the local weather data and just count backwards or forwards. So basically, you don't need to count every single ring, you just need to find a reference point in your local rainfall and then count from there.