What is the necessity of these words "door", "heen" and "toe"? Are they directly translatable? Would "Ze loopt onder de bomen" make sense?
I don't know the official term of those words (adpositions?), but in Dutch they are very often used to sort of strengthen or give emphasis to prepositions, in the above sentence onder. Door in that sentence is like the neath in underneath. While it's not incorrect Dutch to say Ze loopt onder de bomen, a native speaker is more likely to add door. This site, https://en.wiktionary.org/wiki/Category:Dutch_circumpositions, lists such frequently used words and you can click on the individual words to get more information.
The 'compound prepositions' onderdoor and overheen are called "circumpositions" in the modern grammatical model.
The latter parts of the prepositions (here being door and heen) give "position" to the former parts of the prepositions (being onder and over). By including door, we know that she is walking under a few trees and through to the other side. Without door, there is an implication that she is waking under the trees generally, and possibly for a while (as in a forest).
@Dutchesse722 "While it's not incorrect Dutch to say Ze loopt onder de bomen, a native speaker is more likely to add door."
But doesn't that indicate that ... EN: "She walks (or passes) under the trees -- and emerges into the open." DE: Sie läuft unter DIE Bäume (accusative=motion [across a border])
NL: "Ze loopt onder de bomen." EN: "She walks under (the canopy of) the trees -- and stays under them." DE: Sie läuft unter DEN Bäumen (dative= location [or here motion withing an area])
Sorry to be so pedantic.
I don't find you pedantic at all -what's more: I think your explanation is quite clear.
I struggled with this too and asked my Dutch neighbour. He tells me that door and heen are only used for emphasis. He says you wouldn't say "ik loop over de straat heen" as that is a fairly standard and straightforward thing to do but you might say "ik loop over de muur heen" as it's difficult and not something you do often
Duolingo gave me the following as the correct answer "She walks through under the trees." That is not English!
I think the problem is not the preposition, but "tree" instead of "trees".
This reminds me of the Afrikaans (South African Dutch) sentence that can't be translated into English: Hy het onderdeur die tafel gekruip. Literally, 'He crawled under through the table'. If you say he crawled under the table, it sounds as if he stayed there, and if you say that he crawled through the table it sounds as if he has paranormal powers.