A friend's 11 year-old boy fell out of a tree when a skinny branch he was hanging on broke, he survived but suffered permanent brain injuries.
I free climbed trees starting when I was 7 or 8, by the time I was 12 I was climbing into the tops of tall trees, moving from one leader to another at height etc. All of my friends were, none of us had any serious falls.
Kids do a lot of things outside of their parents control (at least we used to) best a parent can do is teach their kid solid free climbing principles like 3 points of contact at all times, feet and hands on branches close to the trunk, stay off dead branches etc.
The desire to climb trees is deeply rooted in our evolutionary history, I've seen 3-year olds grab around a small trunk and start shinnying right up. There's no way any parent is going to stop a kid who wants to climb unless they hover over them 24/7.
-moss