When you travel with a baby or small child under 2, you can get a lap ticket.
Mom or dad just holds the baby or the airline will even issue you a bassinet to be placed at your feet if yours is a small enough child under a certain length and weight. These tickets for your little ones are generally 'free' for domestic tickets but on international fares, you pay some money.
In fact, internationally, it's 10% of the adult full fare in many cases. And even if you book award travel, say, 50k per person to sit in econ on tickets from BOS-AMS that would have cost $650-700 each, they may nail ya up to $250 for the baby because the adult full fare ticket would have been around $2,500! Like OMG.
However, if you are flying on miles, I have potentially found some ways around this and it does take a savvy (probably kid-friendly) CSR...
Remind them your ticket was an award ticket and suggest they take the 10% off its value not a full fare one.
NWA had charged me the big numbers a few times til I ran into this by accident and ended up having a CSR do it for me. The trick is that you need to set up the baby ticket much closer to the time of travel--like up to a day in advance to make it work. Of course, as you get closer the price for that full fare ticket goes up, but on the other hand, if you do it right before travel time, you may get lucky. But I find by doing so, you make THEM nervous that they find out you need a ticket to take your innocent child! You can hear the CSR frantically typing looking for ways to save you money too. Plus, it can often be a paper ticket due to its class, and so they find themselves trying to work magic because you want it issued so close to travel date time and it can no longer be mailed.
There's no link here or way to 100% nail it all the time, but I just thought I'd post this to see what others who have children have run into. Airlines can also issue these tickets at the gate but that may take up your time standing there at the counter. Of course if the baby is cute, it may help.
Incidentally, BOS-AUA cost only $12.95 for the baby on AA. I guess they have different rules. Sadly, no airline is giving your child 10% of the miles on the ones where you do need to pony up the big bucks, but at least you arent buying him or her a whole seat just to sit there and not even enjoy the movie

MM