All three are from Old English.
Dog is found in Old English, but it’s ultimate origin is a mystery. No one can figure out where it comes from. Cognates in other languages appear to be borrowings from English. (I really need to write this one up for the Big List.)
Tree is also from Old English (treow), but unlike dog it has lots of cognates in other Germanic languages. So no big mystery with this one; the etymology is quite straightforward.
Bird is from the Old English bridd and bird in Northumbrian, meaning “nestling, young bird.” Simple metathesis there with the switch of the <i> and <r>. But like dog, it has no cognates in other Germanic languages.