Not sure how to answer the first part of your question, about the role of forests in maintaining ecological balance since forests are very dynamic and complex ecosystems themselves. About the second part, the role of forest trees in the carbon cycle, among the terrestrial ecosystems, forests are the major carbon sinks at the global level. For example, forests in Europe are responsible for absorbing 155 Mt of CO2 per year, a rough 10% of gross GHG emissions (Forest Europe, 2020).
I am sure you can find more information by looking at documents such as the Global Forest Resources Assessment by FAO ( https://www.fao.org/forest-resources-assessment/en/)