This is how I see the different parts looking:
The organic parts, such as the trees and leaves will follow a classical Interpritation, as they must not look sci-fi but earthly. The tree will be large and knotted and air roots and plants rising from the lake that the tree sits in.
The container pods:
A mix between the organic and the mechanical, the pods resemble that of an organic nut. There lower half even looks like one while the top half looks like a machine version of it. It keeps with the white aesthetic and gives the impression of feeding into it with wires and the such. There is a slit along the bottom that can open and close.
A large glass cylinder that feeds into the ship. Wires connect the rings (and hence the pods) inside to the ship base. Roots entwine around the outside of the cylinder and a few roots drop outside the field.
The ship and parts connected to have a clean atheistic, as the wires and mechanical parts are concealed underneath, away from the travellers onboard. The top side of the ship will house a small lake that the tree sits in (hydroponics) and path ways up to the habitat rings. It will look very clean and minimalistic other then the trees and environments.
The underside will be slightly more industrial, with storage facilities and the needed equipment to keep the ship in the air. Roots from the tree above will entwine around these structures.
The beams that project the force field around the treeship are produced by two machines underneath the tree. The beams are visible but shoot down, so as to stay away from the people above. The beams will mainly be light focused to create a union between them and the see-through barrier that encases the ship.
This is how I envision each part of the ship and how I plan to style them during the modelling stage.