So I listened to a presentation by the leading expert in pediatric sports medicine who was also the team physician for Team USA in the Sydney Olympics. Here are his thoughts on children training.
Training is possible at age 7-8 when we've learned to balance. However it is appropriate for very few children.
Training for children before puberty works best with sports specific strength training in addition to cardio training.
The 13-16 year age is the most concerning age for strength training because children are at the peak velocity of growth. And if you are going through a big growth spurt, your bones may "outgrow" your tendons which put you at a greater risk for injury.
And once you are 16 and have likely reached full puberty, you have the hormones on board to maximize strength training, and vertical growth has started to slow down making you less likely to injure yourself.
And of course, we are all different, so each of the above are subject to variation.