Parse Trees and Operator Precedence

Question

I have the following basic BNF grammar:

<A>   -->  <id>  = <E>
<id>  -->  A | B | C | D | F
<E> --> <E> + <T> | <T>
<T> --> <T> * <F> | <F>
<F> --> <E> | <id>

Operator (*) should have higher precedence over (+). However, see this statement:

A = A + B + C + D * F

The leftmost derivation (and parse tree) indicate the opposite! i.e., (+) has higher precedence than (*) since it will show up lower in the parse tree!

What am I missing here?

Answer 1

Due to the path

<E>  ->  <T>  ->  <F>  ->  <id>,

you can completely circumvent the "check" for whether there are *.

Think about it in a bottom up manner: only if there's no * nearby can you parse a +.