My advice would be to hunt down a violin teacher. A good teacher of young students can help you fit the instrument to the child properly and they usually offer a trade-up service to provide progressively larger violins as the child grows. Kids can out grow fiddles as fast as they outgrow shoes! A teacher is also your first line of defense against wasting money on junk.
But to answer you question directly, violins are made in sizes from 4/4 (full size concert instruments) all the way down to 1/32 (very tiny, almost toy-like fiddles). But a 1/2 size violin, for instance, is not one-half the length of a full size. It’s a graduated scale.
Rent. Fractionals can be hard to move at times. When renting from a good shop, you can trade up to the next size, and apply most, if not all of the fees you’ve paid to the next instrument.
Brands are not so important as the person who hands it off to you. It’s all about the setup. A skilled luthier will do a better job at setting it up than a general music store clerk.