Under Section 62 of the Act (below), the owners Corporation has a legal duty to maintain and repair common property. You could argue that while the tree belongs to the neighbour, the airspace containing the offending branches are common property.
The owners corp (not the EC) can only decide not to maintain and repair CP if they pass a speial resolution (more than 75 percent of votes at a general meeting) and if doing so would not affect the appearance of the block.
In short, if I am right about the airspace being common property, the EC ihas no right to refuse, regardless of how few residents would be affected. Threaten to run them through Fair Trading and NCAT whioch will cost the building even more than a tree surgeon.
62 What are the duties of an owners corporation to maintain and repair property?
(1) An owners corporation must properly maintain and keep in a state of good and serviceable repair the common property and any personal property vested in the owners corporation.
(2) An owners corporation must renew or replace any fixtures or fittings comprised in the common property and any personal property vested in the owners corporation.
(3) This clause does not apply to a particular item of property if the owners corporation determines by special resolution that:
(a) it is inappropriate to maintain, renew, replace or repair the property, and
(b) its decision will not affect the safety of any building, structure or common property in the strata scheme or detract from the appearance of any property in the strata scheme.