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 the client should do it, attach one tag to the immediate previous and one to the next level up

yes, otherwise it is a mess, trees get nasty when they get long, losing context 
 just to be clear what this is, is a causality chain... the multiple messages in the same segment are naturally not sequential, but everything else should be, and linking to the tree upwards as well as the previous branch ensures that timestamps don't rule the rendering order, because replies are causal, not sequential

if you are not making a tree then you trust the timestamps but because they are applied by the client you have a problem where users can mangle things with slow clocks, and adding this causal chain ensures that the client has multiple options to delineate sequence and association