[teataster]

There is something I am missing here.

How does your mental model change that much from where the operator goes?

Can't you put node to right of a column of children? Like you would do on a piecewise function.

I am dislexic, maybe that's why I do not see your point.

[Jtsummers]

Prefix and postfix notations permit a great distance between the operands and the operators. If done in a disciplined fashion so that they're merely swapped (at least in the majority of cases), then sure, it's not too different. But consider this: (¬ used for unary negation)

  b ¬ b 2 ^ 4 a c * * - √ + 2 a * /

The two expressions being divided are quite large, so there's no easy way to just move the division one token to the right, it goes all the way to the end. Which two things are being divided at a glance, just point them out. I've even selected a more familiar expression for this exercise that should make it easier. But what about an expression people are less familiar with? Which thing are we calculating the square root of? This is what RPN looks like when entered into a calculator, which is handy and quick for computations (I mean, I have an HP-32S II at my desk next to my laptop, I like it). But even the HP-48G with its graphical equation editor didn't force you to think like this, algebraic notation provides locality of information that makes it much more comprehensible without overburdening your working memory.

And once you start adding parentheses or indentation to convey that information, you're on your way back to a tweaked infix notation. The same for prefix notations. If you can see past the parentheses of Lisp/Scheme it's not too hard to see what's being expressed (using the same symbols as above):

  (/ (+ (¬ b)
        (√ (- (^ b 2)
              (* 4 a c))))
     (* 2 a))

This is more comprehensible, but it's not a strict prefix notation anymore, we've used parentheses and indentation in order to convey information about the grouping. I've even borrowed Lisp's use of variable numbers of arguments to simplify *. If asked, "What is being divided by what", you can, rather quickly, identify the divisor and dividend, at least point them out if not express totally what they do. But a straight prefix notation or postfix notation makes that task very difficult.

You could start naming subexpressions, but then you get into one of the two hard problems in computer science (naming things).

And then we get to the other things people want to do with mathematical expressions, which is largely symbolic manipulation. The present infix notation (or a highly demarcated prefix/postfix notation like the Lisp-y one above) is much better for this task than a strict prefix/postfix notation.