Trees are without a doubt the best carbon capture technology in the world. When they perform photosynthesis, they pull carbon dioxide out of the air, bind it up in sugar, and release oxygen. Trees use sugar to build wood, branches, and roots. Wood is an incredible carbon sink because it is made entirely of carbon, it lasts for years as a standing tree, and takes years to break down after the tree dies. While trees mainly store carbon, they do release some carbon, such as when their leaves decompose, or their roots burn sugar to capture nutrients and water.
Let's look at a real example, a white oak can live for 200 years; all that time it is pulling carbon out of the air and storing it. After several anthracnose outbreaks the tree dies, but it takes decades for the tree to rot. While it is slowly breaking down, the rotten tree is still keeping carbon out of the atmosphere.
Forests capture and store different amounts of carbon at different speeds depending on the average age of the trees in the stand and the number of trees in the stand. Young forests have many trees and are excellent at capturing carbon. Young trees grow quickly and are able pull in carbon rapidly. Not every small sapling becomes a large tree due to competition for light, resources, and growing space, but when they die and decompose little carbon is released. The trees that remain continue to grow and sequester more carbon as the forest matures.
Established or mature forests are made up of "middle-aged trees", which are medium to large, healthy, and have a large root system. Middle-aged trees grow slower than young trees, but the amount of carbon captured and stored is relatively greater. Some of the larger trees occasionally die, but they are quickly replaced by younger trees who take advantage of the new space. Since more trees are growing compared to those that are dying, the overall net productivity (how many trees grow versus how many die) is positive and carbon capture is enhanced.
Old-growth forests have a more fixed, or less dynamic, carbon cycle within live and dead trees and the soil. In old growth forests, large trees dominate by shading out small saplings, so recruitment of young trees and net productivity is zero. Still, the carbon is well contained within the big trees, slowly rotting logs, thick leaf litter and soil. Large individual trees may take up as much carbon as an individual middle-age tree, but since there are fewer trees in an old growth stand, the total additional carbon capture is often lower.
The carbon that is sequestered in forests comes in many forms. For example, forest soils contain plant roots, leaf litter, and other dissolved organic material. The amount of carbon stored in forest soils is variable, and how much carbon soil can sequester is dependent on many local factors like local geology, soil type, and vegetation. In some forests, like in Canada by the tundra, the soil holds more carbon than the trees, but in other forests, like the rain forest, the soil holds relatively little carbon and the trees store more carbon. This is because some soil types, like clay soils, can bind up a large amount of carbon, whereas sandy soils are not able to bind much carbon. Soils with more organic material (bits of wood, decaying leaves, or dead creatures) can store more carbon because organic material easily binds loose carbon molecules and the organic material itself is stored carbon. Soils that are frozen for a good part of the year or have a high-water table can also store large amounts of carbon because decomposition is slow.