It depends how tall the 11 year old is. To see if the bat is the right size for you, you should stand up straight with your arms by your side. Stand the bat up next to you. If the bat is the right size it will be level with your wrist. You should be able to hold your bat out with your weak hand without it shaking. If you can't do this, the bat is too heavy.
Junior's bat are of Size H. Means under aged players can use this size of Bat
34 inch marucci
I would say he/she should use the size they are most comfortable with. If I were you I would make sure that the bat they chose is not to heavy for them and not to bulky. It really depends on how big you're six year old is.
31 inches and 19 or 20 oz
he uses a 43 inch bat