The answer to your question is both yes and no.
There are some trees and plants that do better in warmer environments, some do better in cooler ones - hence the diversity across the different climate zones.
As the planet warms some species will benefit and for the time being the overall effect is a net benefit. Crops have done better overall in recent decades but an optimum temperature has now been reached, any warmer and crop production will start to fall, slowly to start with then at a rate of 10% of global production for each 1°C temperature rise.
The things that trees need to grow are water, sunlight and carbon dioxide. There's more CO2 that they need so reducing or increasing levels of this gas will have little effect. More important is water and sunlight and levels of both of these are affected by changing temperatures.
As the world warms or cools it causes a range of shifts in weather patterns - primarily driven by ocean temperatures and wind generation radiating outwards from the Equatorial regions.
Where this leads to warm or hot regions with high rainfall then trees and plants thrive but as the temperatures rise more rainfall is needed. If it's not forthcoming you get areas of desert or scrub in which very little can grow.
In short, there's more to tree rings than just warmer or cooler temps but overall trees and plants will do better in warmer climates. I've simplified the answer but I hope it makes sense.
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RE: YOUR ADDED DETAILS
In terms of global crop production temperatures are more or less at their optimum, any further increase in temperatures will see crop production fall. If you think about the hot climate zones, they're limited in what they can grow. One of the main staples in these areas is maize and because of the climate they're able to harvest two crops a year - early mazie and late maize. In recent years the late maize crop has struggled and growth has declined significantly. I was in Africa last year and this was one of the things I studied, here's some photos:
The first two pics show what the late maize crop should look like early in the season, the second two are typical of the situation across large parts of Africa and Asia.
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http://profend.com/ya/Image3189.jpg
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http://profend.com/ya/Image2680.jpg
The green crops had been watered but shortly before we arrived the well had dried up. Normally there would have been 10 wells in the area but now there was only one - at the local school about three miles away.
In terms of a single optimum temperature, there isn't one. What's good for one thing isn't necessarily good for something else. The planet is in a constant state of flux with the temp always going up or down. Ordinarily, these changes occur slowly and species are, by and large, able to adapt and evolve so as to keep abreast of the changes. Things are different now and the rate of temperature change is too fast for some species to keep up with.
Your observation about the Equatorial regions being lush and green is spot on. As the planet warms, and subject to the availability of sufficent rainfall, this type of vegetation expands it's coverage (ignoring the fact that it usually gets cut down for grazing, firewood or commercial timber). In times when the Earth is much hotter than it currently is, much of the land is covered by tropical forests.