A manager at the daycare, which employs about 10 staff, travelled to hospital with the injured children, while ambulance crews inside checked all children who were outside when the tree fell.
One child was critically injured and another was seriously hurt.
A third child had moderate injuries.
The three were taken by ambulance to Starship and a fourth child with minor injuries was taken to hospital by their parents.
In a statement released through Acumen, Discoveries Educare said it apologised to every family with a child in their care.
"Discoveries Educare apologises to all our families, and are extremely grateful for the support and understanding they have shown us during this difficult time.
"We have been actively supporting the affected children and their families. Our staff have been supporting these families through his difficult time, and we are thankful it appears there will be no lasting effects for these children."
The daycare was "cooperating fully" with the Ministry of Education and Worksafe regarding the incident and approval was given for the Centre to reopen on Friday.
"We take our responsibility caring for our children extremely seriously. We pride ourselves on our safety record, and all our centres comply with appropriate health and safety regulations."
Discoveries Educare owner Ajit Singh said he was shocked by the incident, and hadn't been aware of any previous concerns about the tree.
"I think this weather for the last two days has been so windy," he said on Tuesday.