In 2006, Scotland banned smoking in public places - pubs, workplaces etc.
You might think that this doesn't do much for children because (1) they aren't in pubs and (2) they aren't at work. Not so, as this paper shows. It's a time series analysis, so not proof beyond all doubt. But it's hard to think of anything else to explain the picture. Notice that the the paper shows data from 2000, and a good data run after the legislation. This is important - beware a time series which doesn't give you a good long data run before the period of interest.
Hospital admissions could in theory go down if children are dying more before reaching hospital - but there's no reason to think that's what is happening here.
The tobacco industry argued that a ban on smoking in pubs would mean people going home to smoke and so making things worse for children. They were wrong.