You're almost right, but not entirely.
Ett barn, barnet = a child, the child
Barn, barnen = children, the children
The word 'Trädgården' looks like it was a word for a garden with fruit trees in it. I find that confusing! :|
Fruit trees used to be very common, everybody with their own house with garden, used to have apple treas, or cherries. But it does not need to be fruit. Our gardens have a lot of trees and scrubberies
Oh, that's interesting! And of course, we also have many trees in our gardens in Germany! In the garden of the house I live in are several (smaller) trees.
Actually, it is not entirely silent, but very very soft, more like an extension of the "r" . The 'd' can be made heard more, if we want to clearify what we say, but we usually soften it down. Edit: I just realized there are two soft d in this word. But what I wrote counts for both, more or less. The 'd' in 'träd' is kind of 'silent', extending the vowel 'ä' a bit, and disappearing becasue of the hard 'g'-sound. And then the second 'd', after 'r' is very soft as i wrote above...
To me, the first D has always been silent, the Ä short and the ending shortened, almost as if it were spelled "Träggår'n". But perhaps that's just my Stockholmer way of saying it.
I agree, when 'trädgården' is pronounced very fast/unemphasized, so that 'e' in the ending is not pronounced, then the last 'd' (in gården) is also dropped (silent). A funny thing is that when the noun gården stand alone (not being in a compound word) pronouncing without 'd' (går'n) will sound more dialectal/country-style/, so when in Stockholm I would pronounce -de- as well, even if very soft. On the other hand, the whole 'compound-noun' trädgården feels too long for that and I will sound more like "trä'gå'n"