World Trees are massive, advanced Elder Tech biological constructs that replicate Dyson Sphere technology in a living framework. They replicate many of the existing properties of planets, but on a stellar scale.
Following is a short summary of simulated or replicated planetary systems.
Despite the word "tree", World Trees have only a cursory resemblance to Terran lifeforms. They make heavy use of Carbon Allotropes, creating much stronger and useful structures than out of wood. The primary support is graphene layers, grown in "branches", which are then covered by something analogous to a planetary crust. Viewed from a distance, World Trees appear as tightly woven webs, with only small gaps and openings. Note that these terms are relative - the smallest branch known is still over one million kilometers wide.
Other allotropes, such as lonsdaleite, are used where non-tensile strength is required. Use of non-carbon is minimized. The seed source of World Tree material is unknown, but is suspected to be harvested nebulae.
World Trees spin, but at a rate which would provide only small fraction of planetary gravity. Through a currently not fully understood mechanism, World Trees convert and amplify this into a simulated gravity effect of roughly .78G.
Polar regions are left uninhabitable, with no gravity or atmosphere, but the majority of the World Tree maintains comfortable levels.
Simulating biological processes, the World Tree regulates Oxygen, Nitrogen, and Carbon Dioxide on a massive level to maintain a fairly normal mixture (there is evidence that other World Trees exist for different biological requirements). This is supplemented by actual flora and fauna living on the interior of the tree.
Gravity, combined with walls on the branch sides, is used to contain atmosphere. Trace losses are replenishes by new material transmuted from the solar wind.
Over long periods of times, branches move together or apart, with some process to regulate crusts. This appears to be an attempt to simulate plate tectonics, and the end result is very similar.
Instead of subduction, when two branches merge the material from one is instead harvested and extruded elsewhere, to provide "fresh" crust material where needed.
Power & Day/Night cycle
Giant "leaves" extend out into the areas between branches. These harvest solar energy, as well as the solar wind for raw materials. A second set of leaves is used to create a day/night cycle, slowly closing over populated areas and then re-opening.
Excess power is provided by special "trees" that grow on the inside, extruding from the World Tree branches through the crust. Power is generally sufficient to maintain a Tier 1 civilization.
Reclamation & Pollutants
The World Tree actively filters its water systems and atmospheres to maintain the original levels of trace elements. Some storage is assumed, but excess trace elements are incorporated back into fresh crust material, making them - in some cases literally - gold mines of raw material.
The World Tree manipulates its leaves to maintain position around the star. By trapping the wind in one direction but venting in the other, this produces a small but significant thrust that can be used to counter impacts or other accelerations.