Forests play an important role in models predicting how increasing CO₂ will affect climate, because trees remove carbon from the atmosphere and lock it up in wood. However forests themselves are both vulnerable to the effects of changing climate, and benefit from increased CO₂ in the atmosphere (if more is available, trees can be more economical in the amount of water they need to absorb it by photosynthesis). Disentangling this role is difficult, because future changes may change not only how well a forest grows, but also the species of trees within it.
A recent paper by several authors associated with the Tree-Ring Lab reconstructs past North American forest growth from tree-ring data, then predicts the changes under future climate change without assuming that the type of forest in a given place will stay the same. If there is a forest perhaps many hundreds of miles from that place which is now growing under the expected future conditions, this is a much better pediction of how the forest will behave in future. This approach can also include many of the effects of increasing CO₂ by matching the future conditions to present-day forests that experience higher rainfall (since trees in the future forest will use water more eficiently). At the scale of the North American continent, it predicts a worrying overall decline in forest growth (even if trees can get by with much less water than at present because of increases in CO₂). A hope from previous models had been that in future forests in Canada and Alaska would grow better in response to warmer temperatures, removing CO₂ from the atmosphere and slowing the overall rate of change, but this more realistic study fails to find that effect.
Charney, N. D., Babst, F., Poulter, B., Record, S., Trouet, V. M., Frank, D., Enquist, B. J. and Evans, M. E. K. (2016), Observed forest sensitivity to climate implies large changes in 21st century North American forest growth. Ecol Lett. doi:10.1111/ele.12650