It's actually Czech feminine first name, too. More common version is Doubravka. And yes, it's related to oaks. But I'm not sure if it means "oak forest" only, because Doubrava is a Czech river, too. There the ending -ava is one of the most common endings of Czech river names (as in Vltava, Morava, Sázava...), thought to be related to the Celtic Avon.