Time to play a bit. I just learned about rapidly-exploring random trees while reading about motion planning. They look kind of like drainage networks so I had to try them out. I created a disc (a perfectly round island, if you will) and then generated a tree in the interior. Here are a couple of example results (click to enlarge):
The gist of the algorithm is that you generate a sequence of random points distributed uniformly over the interior. For each point, find the closest point on the tree and create a new branch from there, headed toward the random query point. You don't go all the way to the query point, though; just a small step. The net result of this is that the tree tends to head toward open areas.
I've shaded the tree edges as a function of how many other edges empty into them, so you can see the big rivers are darker than their tributaries.