I will apologize up front if any of this below is redundant for you folks. If you're already familiar with dynamic-ropes and releasable-hitches, then skip, or treat this as a refresher...
Since we're discussing falls & fall-protection, It must be said that folks should have an understanding of the importance of "dynamic" ropes, which are designed to stretch, and thus offer some shock-absorption. Us Cavers, and Arborists don't use these
- we use stiff "Static" ropes that don't stretch, so that we can climb them easier.
Rock climbers use dynamic ropes, as they're typically climbing the rock, and they plan on taking falls onto the rope at some point. This is what you want to be looking for. Since you're talking about using these in trees, and I presume you're using a stick-climbing system, I wouldn't geek-out too much on choosing a dynamic rope. Go to HowNot2.com and buy some by-the-foot.
...But before you do, you need to think about what you'll do
AFTER they fall... There's no use in catching your child
(or yourself) with a rope, if you have no plan after that, than just dangling there until a hungry bear comes along...
Lookup releasable hitches - Munter hitch would be my recommendation - and practice using it.
Make sure that you bought & brought a long enough section of rope, that you can lower that person to the ground. Having to cut them free won't help a whole lot... After all, it's not the fall, it's the landing that'll get ya'...

Climb safe.