Papaya here just grows, and they get pretty tall too. Sometimes it can get interesting when you have a Papaya that is 10 meters tall (at least it seems that tall!) and you are shaking it to make a fruit drop.  
 
  Lots of roots - and they go out some distance. I have never really noticed anything being planted with them, they just come up volunteer for us, and of course, we are in the tropics, so they don't get killed off by freezing weather.