I prefer a .22. Don't own a .17, but that is a good choice. 12 gauge is fine except you are over gunned which is way better than under gunned. A .410 for the child may be a good choice.
Keep in mind the shot is normally upwards. Unless the tree is offering a backstop, the rifle bullet will travel a long distance following a miss.
I used a Ruger 10/22 with hollow points. No longer hunt them as I do not eat them. Sure is a fun hunt though.