Parenthesis

To a computer, there is no difference between the expressions (((x)+(y))(t)) and (x+y)t; but for a human, the latter is easier to read. When expressions are automatically generated for humans, they should contain the minimal number of parentheses for the simplest perception. We assume that expressions consist of only two operations: addition ($+$) and multiplication (juxtaposition), and that these operations act only on single lowercase letter variables. Consider the grammar for an expression E:

E : P | P '+' E
P : F | F  P
F : V | '(' E ')'
V : 'a' | 'b' | .. | 'z'

The addition ($+$, as in x+y) and multiplication (juxtaposition, as in $xy$) operators are associative: x+(y+z)=(x+y)+z=x+y+z and x(yz)=(xy)z=xyz. Commutativity and distributivity of these operations should not be assumed. Parentheses have the highest precedence, followed by multiplication and then addition.