Ref discussion at CA. I use simulated monthly temperature record, data here . Simulated ring width response is here. Correlations with monthly temps are here. But as Willis suggested, we can find other time series that give exactly the same correlations.
See e.g. simul1. and simul2. Try with Matlab,

s=corrcoef([originalrw temperature]);s(2:end,1)
s=corrcoef([simulrw1 temperature]);s(2:end,1)
s=corrcoef([simulrw2 temperature]);s(2:end,1)

rw1 and rw2 are obtained using different inital guess (white noise and red noise). Quite different results,