There's a different nuance here though. Your sentence would be:
공원에 나무 없어 - There are no trees in the park. Whereas this sentence is - There aren't ANY trees in the park.
They both mean the same thing, but the translations have different nuances.
You can find the explanations for the particle 도 here:
This is not grammatically equivalent, in your sentence, "the park" is the subject of the sentence. The original Korean sentence uses the locative particle 에 as in 공원에, not the subject particle 이 as in공원이.