In the first instance you need to speak to the Strata Manager to confirm that the courtyard is part of the Lot, and that the tree there is above the stratum of that Lot as you suggest.
If the above are confirmed, then you or your Strata Manager need to contact the Local Council’s Tree Preservation Officer to ascertain how much of the tree may be lopped and/or under what circumstances it may be removed entirely.
It was actually the responsibility of the Lot Owner to keep their tree regularly trimmed so that it’s kept within the stratum of their Lot, so again after confirmation as per the above, as that trimming hasn’t been regularly done and as the tree is now beyond that stratum, and as it’s in close proximity to the building, and as the external structure (i.e. the walls) of that building is Common Property, then subject only to Council’s advice the Owners Corporation can, at its expense, resolve to do whatever it chooses to with the tree; with or without the Lot Owner’s consent.
As for the Strata Manger – he’s employed by the Owners Corporation, that by the way is made-up of all Owners, and therefore his main responsibilities are to provide advice to the Owners Corporation to ensure that it manages the Property in compliance with the Act, and to do whatever it properly instructs.
So you and the other Owners don’t have to “present a case” to the Strata Manager with regard to the tree or indeed anything else, but rather, have him do some homework for you as I’ve suggested, and if everything is confirmed in terms of the ownership of the courtyard, about the height of the tree relative to the stratum, and with Council’s requirements, then you and like-minded Owners need to instruct him to obtain quotations for the tree to be lopped/removed, and to then submit to him an Item for inclusion on the Agenda for the next AGM to the effect that the Owners Corporation resolves (by a simple majority) to have the works undertaken.
In conclusion, I fully understand that Strata Owners (elderly or otherwise) can sometimes be influenced or even intimidated by their Strata Manager, and as hard as it may be in those circumstances, you and like-minded individuals need to make it clear to such personalities, in writing if that’s less confronting, that they’re your contractor; not your master!