You will see American children in suits, ties and even tuxedos - they appear to dress up more and need this for School Proms - I would suggest shirt and tie, perhaps their school blazer or buy them their first suit..
I wouldn't get too hung up about formal nights. As long as they look smart (by that I mean no jeans, trainers etc) they will be fine. Trousers, shirt, maybe a jacket - no real need for a tie. If you have girls, then a nice dress will be fine. When we were on Liberty towards the end of last year, there was a real mix of tuxes, business suits and people who looked smart without being so formal. Some people dressed their children up too much in my opinion, they're people, not dolls.
Thank you both for your replies.
We sailed on Voyager last summer and my 13yrs old son wore a Tuxedo along with his very very proud father.