I do not know of one - a child would use a seat that could be sold to an adult.
Your best bet is to just look for the cheapest fares. If you can use a discount airline like AirTran or Southwest and can be flexible on the schedule you can find some pretty reasonable fares. The problem is that these airlines have a limited list of destinations, so they may not be practical.
Your options would depend on where you are going to and from. For example a trip from Fargo to Little Rock would not offer you many options.