I looked into such a circumstance months ago, but from the OC point of view.
That said, what I found addresses your problem.
The tall tree was in one lot and did encroach any neighboring property.
I discovered:
If the tree is on a lot’s land ie not common property and is not causing a problem then it is the lot owner who decides if to prune or cut or remove;
In the absence of a direction from the local council to remove the tree, the OC cannot dictate what an owner does with the tree unless the tree is causing problems like shedding leaves on common property;
Many trees in time will cause problems with leaves shedding or expanding roots etc. at that time the OC can demand the lot owner takes action;
If for whatever reason the OC wants pruning cutting or removal tale place asap, they can always offer you to pay part of the cost;
If the tree is designated by council as dangerous, unsuitable for its location etc then if instructed to remove then the owner of the tree, is lot owner concerned, must comply with council directions at his own cost. It is strictly a matter between the lot owner and council; and
If the lot owner wants the tree removed cut it pruned, the questions are (1) is the tree of a type or size that requires council permission before taking action and (2) if council permission is needed, then correspondence is between the lot owner, the strata plan in this case is a bystander, and council and the cost is for account of the lot owner himself.
Hopefully the above clarifies the issue.