Self-nitpick (too late to edit my post above): I used the phrase "special case" wrongly here -- restricting the valid inputs to a problem creates a strictly-no-harder special case, but constraining the valid outputs (as I do here regarding left-deep trees) can sometimes actually make the problem harder -- e.g., integer linear programming is harder than plain "fractional" linear programming.

So it's possible that the full optimisation problem over all join tree shapes is "easy", even though an output-constrained version of it is NP-hard... But I think that's unlikely. Having an NP-hard constrained variant like this strongly suggests that the original problem is itself NP-hard, and I suspect this could be shown by some other reduction.

with selectivity a function of only the two immediate child nodes in the join

This should be "with selectivity a function of the rightmost leaves of the two child subtrees", so that it still makes sense for general ("bushy") join trees. (I know, I'm talking to myself... But I needed to write this down to convince myself that the original unconstrained problem wasn't just the (very easy) minimum spanning tree problem in disguise.)