Count the number of part expressions of an arithmetic expression

The problem:

Define the recursive function numexprs that returns the number of part expressions when given an arithmetic expression. Note that even bool is a part expression.(Individual parentheses are not counted as part expressions.) Evaluate your function on the expression below, and verify that it returns the expected value of 8.

I don't understand how I should define the recursive function. The question states that I should count the number of part expression in a given arithmetic expression, but the example I'm supposed to verify isn't even an arithmetic expression? I feel very confused.

I was thinking of using something similar to this when defining the function:

Solution 1:

Let's provide indices to the different bool's so we can see better what is going on. OK, so you have the expression:

$((bool_1 \land (bool_2 \lor bool_3)) \land (\neg bool_4))$

The $8$ subexpressions of this expression are:

$bool_1$

$bool_2$

$bool_3$

$bool_4$

$(bool_2 \lor bool_3)$

$(bool_1 \land (bool_2 \lor bool_3))$

$(\neg bool_4)$

$((bool_1 \land (bool_2 \lor bool_3)) \land (\neg bool_4))$

OK, now follow the hint left by Mauro in the Comments