Thats also what I have interpreted but has not been accepted for some reason which I do not fully understand. "The tree is also here" implies that there are some other objects there besides the tree (which is what I had assumed when reading the Russian sentence), but I do not know if that is the case in "Здесь тоже дерево".
Yandex translates "the tree is also here" to "Дерево тоже здесь", so perhaps the word order has some significance in Russian.
Why "Tam"? I'm not trying to correct you as my Russian is pretty basic, but the word "there" in "There is also a tree here." doesn't really mean 'there' literally. You can call it a dummy subject or an existential there - or whatever - but "tam" seems to me to be translating it literally as if it really meant there. A similar example is the sentence "It is raining". The word "it" doesn't really mean anything either, and I guess Russian would simply say something like "дождь идёт"