Water, soil nutrients, as well as site disturbances such as erosion severity, windstorms that strip leaves or upset root or stem damage, ice storms, moderate to severe wildfires, thinning of forest stands, insects, disease, etc. could contribute to growth variances. So research that attributes growth change to only one factor is difficult, except under the most controlled conditions. And the climactic and weather changes as a result of increasing CO2 are not felt equally across the earth.
There are tree ring studies that include the effect of CO2. There are issues like tree ring methods are not that good in detecting effects of CO2. Also tree rings miss changes in stand density (so tree growth rates may not change but you may have less or more trees in a stand).
A not anymore so recent review is the following paper.
Article Response of Forest Trees to Increased Atmospheric CO 2
carbon dioxide concentration increase from 300 to 400 ppm,means the increase 1/10 000 and increase of Total Solar Irradiance over the same period on 1 Watt/m², or 1/1365
Tree ring width is strongly affected by wind sway so changes in the local windiness would likely cover any other changes. Boreal trees furthermore react to wind sway during dormancy so heavy snow load during winter affects the stem radial growth the following years.
CO2 should have an indirect effect, possibly resulting in larger tree crowns or slightly higher trees, but that in itself doesn't necessarily result in wider annual rings unless it also result in more wind sway.