Case 1: Size Based on Age and Height (Conventional Method)
A properly fitting bike allows a comfortable and safe sitting position for children as well as teenagers. The size of a child’s bike is usually based on the wheel diameter. In adults, this is based on the frame size or seat tube length. Proper bike fit allows the muscles can deliver optimal strength. This is less tiring and avoid overloading the knees. Because children have less experience and balance, they are more susceptible to falls. However, the habit of cycling is taught at a young age and a seat tailored to child’s height ensures optimum comfort and encourages them to ride on their bicycle. All the while teaching them how to ride more intuitively. So how does one choose the right size of a child’s bicycle? Here is a table for a general rule.
|Age||Body Length||Bike Size|
|2-4 year||87-103 cm||12 inch|
|3-6 year||96-117 cm||14 inch|
|5-7 year||110-123 cm||16 inch|
|6-9 year||117-135 cm||18 inch|
|7-10 year||123-140 cm||20 inch|
|8-11 year||129-146 cm||22 inch|
|10-14 year||140-164 cm||24 inch|
|> 13 year||160-176 cm||26 inch|
Here is a graphic that describes the size for kids better. (I do not own the rights to this graphics, credit goes to http://www.wiggle.co.uk/)
Case 2: Size Based on Inseam and Height (More Advanced)
The size chart for children is based on the age and height. Since not every child is equally tall at a certain age, it is better to choose the bike size based on body length. A study  in which young children between 7 and 14 years shows that choosing a bike just based on the bike size is a not the proper way. Along with the bike size, the most influential parameter for a sitting position is the inseam. This is significantly different for children even with the same height. On the basis of the body size of 1,400 youth one arrives at the conclusion summarized in the following table, which takes into account the growth:
|Body Length||Inseam||Bike Size|
|120-140 cm||< 52 cm||16 inch|
|52-74 cm||20 inch|
|> 74 cm||24 inch|
|141-160 cm||< 62 cm||20 inch|
|62-80 cm||24 inch|
|> 80 cm||26 inch|
|> 160 cm||< 74 cm||24 inch|
|>74 cm||26 inch|
The study was conducted over 1400 young folks. 9 body measurements were taken which in the end determined the size of the bike. With these data, one could find out which size fits the children for any body length. The only thing you need to adjust is the seat and handle bars. Criterion of a good seat height is the distance between the pedal and the saddle corresponding to 88.3% percent of the inseam. Children between 120 cm and 140 cm tall fit so almost on a 20 inch bike. For children between 140 cm and 160 cm the average bike size is 24 inches. For teens bigger than 160 cm, the best fitting bike would be a bike with wheel size of 26 inches.
Although most would end up preferring to go for the standard and easy to use first case, some like to go beyond by paying close attention to detail and choosing the bike that would perfectly suit them. This would require more hard work, but in the end would make your purchase all the more enjoyable. you can always check out the comprehensive study referenced below for more information.
Reference  Laios L, Giannatsis I. 2010 Ergonomic evaluation and redesign of children bicycles based on anthropometric data. Appl Ergon 41: 428 – 435.