The 16-year-old was last seen boarding a bus in Aberdeen on 17 November last year.
A member of the public discovered a body on Friday afternoon in woods south of Banchory in Aberdeenshire.
Police Scotland have now confirmed it was that of the missing schoolboy.
In a statement released by police, Liam’s mother Alix said: “While we never gave up hope of finding Liam, we were realistic about the challenges facing us. Our hearts are broken.
“We would like to take this opportunity to thank everyone who has been involved in the search for Liam including members of the public who took it upon themselves to help.
“Thank you also to everyone who has left messages of support which have provided us with comfort at extremely difficult times.
“At this most difficult time of all, we would ask for privacy as we try to come to terms with our enormous loss.”
Specialist resources including a police helicopter, search dogs, and the dive and marine unit had been deployed in the search for Liam.
Police Scotland said there are not believed to be any suspicious circumstances surrounding his death.