Sutherland Shire Council has overturned staff decisions in the first two cases under a new system for handling complaints about troublesome trees.
In one case, a resident carrying her one-year-old grandson narrowly escaped being hit by a large, falling branch.
Under the new system, if a resident's application to remove a tree is refused, a councillor can have the decision reviewed at a full council meeting.
This week's meeting voted 10-4 to allow the removal of trees in Miranda and Gymea Bay.
Greg and Leonie Miller were extremely relieved at the decision regarding a 12 metre high gum on the council strip in front of their home in Bimbadeen Avenue, Miranda.
They thanked Cr Barry Collier for taking the matter to the council floor.
Mr Miller told the meeting they had tried unsuccessfully for more than four years to get permission to remove the tree.
Mr Miller said falling branches had smashed the roof of their home, cut power lines and damaged cars.
"My wife narrowly missed being hit by by large branch while she was holding our one-year-old grandson," he said.
Mr Miller said the tree had not been planted by the council, but by an earlier resident, and offered to pay 100 per cent of the cost of removal.
The staff report, which was considered at the meeting, said the council had been contacted 10 times since 2014 about the tree, which was one of the few remaining in Bimbadeen Avenue.
Deadwooding and minor pruning had been carried out.
"The tree in question is a large, healthy, vigorous tree that makes a very positive contribution to the locality visually and ecologically," the report said.
"The tree is a native species to the east coast of Australia.
"The management of canopy trees can be an emotive issue.
"In order to arrive at consistent, rational outcomes, each request for removal of trees needs to be assessed against a set of accepted and robust criteria.
"In this instance, the tree has been thoroughly investigated and analysed by a number of appropriately qualified council officers with the conclusion being that there is not sufficient justification to remove the tree at this time.
"The minor issues with the tree can be addressed through tree management activities and risk mitigation options."
The council meeting also approved the removal of a tree in the front yard of a home Coopernook Avenue, Gymea Bay.
Cr Kent Johns brought the matter to the council floor for review on behalf of the home owners.
A staff report said the council has been contacted eight times since 2014 about the Angophora costata - Smooth Barked Apple, a local and indigenous species.
Complaints included that roots were lifting the driveway and damaging stormwater pipes and that falling branches were a danger.
The staff report said the tree had been assessed by four different tree management officers over the past five years, and the most recent assessment gave a low risk rating.
"Mitigation measures, such as removal of deadwood, pruning of selected branches, and reduction in the volume of canopy have been approved on several occasions previously," the report said.
"Further mitigation measures can be carried out to further reduce any risk.
"The concerns in relation to the lifting of the driveway and storm water pipes can be resolved through the upgrading of infrastructure by the owner.
"It continues to be the recommendation that the tree be retained due to the absence of justification for its removal and the positive contribution that it makes to the streetscape and local ecology. "
