The force said it was responding to reports of an 11-year-old boy and a 10-year-old - whose gender was not disclosed - at a house on Church Row.
On Saturday afternoon, it confirmed the man had been charged with the attempted murder of the two children, and also one count of assaulting an emergency service worker.
A 13-year-old was also hurt, but the 11-year-old suffered the most serious knife injuries and was hospitalised.
The man was injured himself and was put under police guard in hospital. He has since been released and was being questioned by North Yorkshire Police's major investigation team.
Speaking previously, Senior Investigating Officer, Detective Inspector Steve Menzies said “The incident took place around 8:10am both inside and outside a property in Beckwithshaw which is situated on the main road, the B6161 Otley Road, and is very close to the local primary school.
"I reiterate that this incident is not connected to the school, however I am appealing to anyone who may have been dropping off children at the school, or commuters passing by, to check their vehicle dashcam and report any footage that may have captured any part of the incident prior to the attendance of the emergency services”.
Anyone who witnessed the incident or who has any information, is asked to contact police. Dial 101, press 2 and ask to speak to the Major Investigation Team.
Please quote reference number 12220105995 when passing on any information.