4373 is the AS for the amenity tree industry, but it doesn't specify PPE. Workcover has a controversial draft code of practice for the tree industry which received a lot of flack when it was released because it was based mainly on existing codes for working at heights/IRATA type stuff. Based on the feedback they've received from industry my understanding is they went back to the drawing board.
Climbers are currently exempted from the other related codes for chainsaw operations meaning that they are not required to wear chainsaw protection/chaps when in the tree. Since there currently is no code of practice in place, you could basically do anything you want to in terms of equipment and techniques, but you're still under fairly broad umbrella workcover regulations meaning hardhats, safety boots, hi viz, hearing protection etc. Interestingly, you are really only under these rules if you have workers comp insurance, which you are not required to as a sole trader. There is also no actual requirement for a tree company to have workers comp or public liability currently, though individual councils do sometimes specify that you should have it but they have no real authority to enforce this.
The industry needs a cleanup for sure, but realistically there is no way to enforce any legislation given the nature of the varying daily sites.