TITLE: Quantifiers in a first-order-logic syntax tree
QUESTION [5 upvotes]: I was told that syntax trees of F-O-L formulas are built the same way as propositional ones. Everything seems pretty intuitive to me, but I'm not too sure about quantifiers.
Let's say we want to build a syntax tree based on the following formula:
∀x(S(x)→(¬T(x)))
(where S,T are predicates, x is a variable)
In this case, ∀x bounds to the whole "inner" formula  S(x)→(¬T(x)), so it should be
        ∀x
         |  
         →
       /   \
      S    ¬ 
      |    |
      x    T
           |
           x

Is that correct?  I understand that from a graph perspective, quantifiers-with-a-variable are unary, meaning that they can only have one child. Am I correct in thinking that this child can either be a 
(i) logical connective
(ii) predicate symbol
(iii) another quantifier-with-a-variable
But it cannot be a term (constant, single variable, functional symbol)? 
Here are some visualizations for cases (ii) and (iii) to clarify what I mean:
(iii) formula: ∃x∀y(S(x)∧T(y))                     (ii) formula: ∃x(S(x))

        ∃x                                               ∃x
         |                                                |
        ∀y                                                S
         |                                                |  
         ∧                                                x
       /   \
      S     T 
      |     |
      x     y

These trees should be valid based on my assumption.
Thank you in advance.

REPLY [2 votes]: Correct.  Quantifiers are placed in front of formulas, where a formula is one of the following:

An atomic formula $P(t_1, ... t_n)$ with $P$ a predicate and $t_1, ... t_n$ terms.
Some truth-functional operator applied to other formulas
A quantifier applied to some other formula.

So: you indeed never quantify a term and in the syntactical tree a quantifier has indeed exactly one child.