"My little sister climbs up a tree in a park."
The fun thing about Kanji is that there are sometimes different kinds of kanji for the same word and each meaning is slightly different. For example these three kanji (上る、登る、昇る) have the same reading of のぼる.
上る (To go forward in a upward direction, i.e. stairs)
登る (To climb to a high place, i.e. trees and mountains)
昇る (For natural things to rise, like the sun, moon, or degree of something)
Depending on which kanji you use the meaning will change.
Another example is かう.
飼う is used when you buy a pet.
買う is used when you buy things.
By looking at the kanji you can sometimes see what the subject is.
Yep. Pronunciation (pitch accent) between e.g. 飼う and 買う is different, though. They're only indistinguishable when written with hiragana.
Could a valid translation of the sentence also be as follows?
When I see the sentence in English I feel like one answer is the Duolingo's one which is more like “which tree? – a tree in a park – 公園の木”, and a second one “where did she climb? – in a park – 公園で木…”.
I think it's much more common to put the "....WA" part of the sentence first, but I don't know if it's technically incorrect to not do this. Possibly just strange to the locals?
The english translation is weird here. Shouldn't it be "climbs a park's tree" because of the possessive particle "こうえんの木"?
Also, sometimes they allow omitting "私の" and sometimes not. I got it wrong at first because I thought it was obvious いもうと meant "(my) little sister".