How Does That Tree Grow
In most cases, researchers calculate the correlation between ring widths of a specified tree (or trees) and the instrumental record of the nearest weather station - if there is no suitable station, gridded* temperature data are used. If the correlation coefficient is deemed statistically significant, no matter how low r2 can be, the proxy is said to be calibrated and regarded as a good indicator of temperature ouside of the calibration period. Keep in mind that this is a short and simplified version of the procedure.
This approach has a number of wekanesses. An easy one to spot is the fact that only seldom an instrumental record for the exact tree location is available; the nearest station may not represent well the actual climate in which the tree has grown. Moreover, a researcher can fudge with confidence intervals, gridcell and proxy selection, and algorithms basically at will until the desired result is achieved.
I am not aware of any laboratory/field research that tried to grow trees in strictly controlled conditions in order to examine the relationship between mean temperature and ring widths. Studies and experiments have been performed for the effect of CO2 concentration, exposure to pollutants and other issues, but no temperature.
Of course, the experiment would be long and costly. The outline of the setup goes like this: 2 or better 3 greenhouses fitted with sytems to control light exposure, precipitation, fertilization, humidity, air composition and finally temperature. Hydroponic colture would make some things easier to control compared to soil, at least in a first version of the experiment. One population of plants would be exposed to a certain annual mean temperature, another to higher temperature and the third group to lower temperature. What form the temperature cycle should take has to be defined: it can be a regular wave produced superimposing high and low frequency components, or a random walk centered around a specified mean.
An alternative is to use only one group of plants and subject it to different temperature regimes at different times, but those results would be harder to analyze - and the experiment would need to run for much longer.
Plants of temperate climates need seasonal temperature cycles, and that adds a considerable degree of difficulty to the experiment. Light exposure times must be adjusted to the seasonal variations too. Also daily temperature cycles can be important, and there can be a problem of system response times. It would be hard to simulate frost conditions in a hydroponic system.
Also baseline conditions - what mean temperature, humidity, precipitation amount and regime (in non-hydroponic systems) and light exposure would have to be chosen, together with exact species and variety of plant to study.
Even if most proxyes come from temperate regions, for semplicity the experiment can start with tropical plants, which are accustomed to a nearly-constant temperature and a nearly constant 12-hours light cycle all year long.
The experiment would have to run for at least 10 years at least, in my opinion (so plans must be made so that maintenance and eventual repairs of the equipment will not disrupt the experiment). 5 years may be enough to collect useful data, but I doubt.
Now do not act penurious, I will accept research grant offers with tempered joy.
* Temperature is recorded by a number of station irregularly distributed around the world. Data are then interpolated and published for the coordinates of the intersections of a regular grid (say, 0.5°xo.5°) covering the whole globe.