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Trees that grow in areas with wet/dry cycles also have rings. And since most of the trees from permanently-wet areas also have some kind of annual or semi-annual cycle, I'd guess the ringless ones are a rare exception everywhere.

Palms and Bamboo are technically "very big weeds". They are more related with grasses than with pines and never have rings. Bananas are also just giant herbs.

So Monocots don't have rings. Anything else that is a tree in a tropical forest has rings. It does not matter where they grow. The rings are smaller in slow growing species, and are different structurally in conifers, but this is all.

Wouldn’t a tree without rings still reasonably capture the atmospheric C13:C12 ratio as it grows? Or is the carbon motility within the trunk too high, or the ratio differences too small, to sample a bit near the core and use the ratio there as an age indicator?

Could be (I have no idea), but it sounds like that would be possible to investigate only by complicated laboratory tests. Not a visible change that makes it easy to calculate age by just sawing up a cross section and counting visible rings.