The mother, named locally as store manager Geraldine Newman, and her daughter, Shannon, 11, and son Shane, 6, were discovered at their home on Tuesday morning.
A murder investigation was launched and detectives discovered the body of the suspected killer on a beach in North Wales on Tuesday night.
The body is believed to be that of Ms Newman’s estranged partner, Paul Newman, 42, who is understood to have been living in Normanton.
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A statement from St Wilfrid’s Catholic High School and St Joseph’s Primary School said: “While the police are still working to officially confirm the identities of those involved they are believed to be a Year 7 student at St Wilfrid’s Catholic High School, her brother, a Year 1 pupil at St Joseph’s Catholic Primary School Castleford and their mother.
“Exactly what happened and how this tragedy occurred is subject of an ongoing police investigation. We are unable to provide any further details at the moment but are aware that the police have described it as a domestic-related incident. The students and children have been notified via their form tutors and class teachers this morning. We have ongoing support in place for all our children, students and staff.
“Our prayers are with the family and friends at this very sad time.”
West Yorkshire Police said on Tuesday night that the body of the man, who they have not named, was found at 5pm near cliffs at South Stack, an island off Anglesey.
The beauty spot, which is popular with walkers and bird watchers, is around 160 miles away from Allerton Bywater.
Detective Chief Inspector Warren Stevenson, of West Yorkshire Police Homicide and Major Enquiry Team, said: “The man’s identity is yet to be confirmed but we are liaising closely with our colleagues in North Wales Police as we believe this may be linked to our investigation in Leeds.”
Police were called to the family’s home in Beeston Way shortly before noon and found Ms Newman’s body downstairs.
The bodies of her daughter and son were found upstairs, although police are yet to formally identify the victims.
Mr Stevenson said: “Post-mortem examinations are yet to take place to establish the causes of their deaths but the victims have died in suspicious circumstances and therefore we are treating their deaths as murder.”
According to reports, colleagues of Ms Newman raised the alarm and a note was later found at an address linked to Mr Newman.
The mother was a branch manager for Wilko where she had worked for 23 years and a spokesman said she was a “loyal and enormously respected member of the team”.
The retailer added: “This is tragic news and has devastated everyone who worked with Geraldine as well as impacting team members across the company.
“Our heartfelt thoughts and condolences are with her family, friends and colleagues at this very difficult time and everything is being done to support those team members who are affected by this tragic news.”
On her LinkedIn page, she described her children as “the light of my world”.
Sean Firth, 44, and his 11-year-old daughter Chloe live across the road from the house.
In tears, Chloe described how she played outside in the estate with the girl.
She said: “I just feel really upset and really shocked. I never thought anything like this would really happen.”
She added: “They were really bubbly and nice and kind. She was just my friend.”
Mr Firth said he believed there had been some kind of domestic incident at the house around a year ago when the children’s father was taken away in a police car.
He added: “I only really knew them through passing in the street and playing in the park with my daughter.
“They were nice and friendly, as far as I can make out. My daughter was devastated when she came home from school.”
Anyone with any information, or anyone who witnessed anything in the area of the address leading up to the time the bodies were discovered is asked to contact the Homicide and Major Enquiry Team via 101.
Meanwhile, tributes were paid to Ms Newman - known as Gerry’ - who was store manager at Wilko in Castleford.
A Wilko spokesperson said news of her death would be felt across the company.
The spokesperson said: “This is tragic news and has devastated everyone who worked with Geraldine as well as impacting team members across the company.
“Our heartfelt thoughts and condolences are with her family, friends and colleagues at this very difficult time and everything is being done to support those team members who are affected by this tragic news.
“This is now a matter for ongoing police investigation and we are giving them every support. For this reason, we can’t give any more details at this time.”
Police were called to the terraced property, in the Millenium Village estate, at around 11.50am yesterday after concerns were raised for the occupants.
It is believed Ms Newman’s body was found on the ground floor, while the children were discovered upstairs.
It is not known how they died, although initial inquiries suggested Ms Newman may have been attacked with a blunt object.
Officers cordoned off the street either side of the property, while scene of crime experts carried out investigations inside and their colleagues spoke to neighbours.
Detective Chief Inspector Warren Stevenson, of West Yorkshire Police Homicide and Major Enquiry Team, said: “Post-mortem examinations will take place today to establish the causes of the deaths of the victims who were found at a house in Beeston Way, Allerton Bywater, yesterday morning.
“Work to formally identify the woman and two children is ongoing and we are not yet in a position to publicly confirm their identities, though we hope to be able to do that later today. We are also continuing to liaise with North Wales Police to confirm the identity of the man who was found dead on cliffs at South Stack, Anglesey, yesterday.”
Wakefield Council leader Coun Peter Box said: “This is a terrible tragedy, and our thoughts and sympathies are with family members, friends, and colleagues at this distressing time. We will do all that we can to support both staff and pupils at the school.”