It is mostly a joke for little kids. The whole point is that the word "bark" is being used in two different ways. A tree doesn't really bark like a dog! For some reason, little kids find the idea amusing. Other jokes in a similar style: What did the ocean say to the shore? Nothing, it just waved! Which vegetable is famous for singing? Elvis Parsley!

The fact that both dogs and trees have their bark as a similarity IS the joke. It's a pun, or a play on words.