(1) |

In (1), no node has more than one branch emanating from it. The
nodes in such a simple tree are related to one another by a single
relation, the **dominance** relation. Dominance is a theoretical
primitive; in other words, it is an irreducibly basic notion, comparable
to a mathematical concept like point. Dominance is represented
graphically in terms of top-to-bottom order. That is, if a node A
dominates a node B, A appears above B in the tree. In (1), for
instance, `NP` dominates `N` and `Zelda`, and
`N` dominates `Zelda`. The node that dominates all other
nodes in a tree, and is itself dominated by none, is called the **root
node.**

Does a node A dominate itself? If the answer to this question is
defined to be yes, then the dominance relation is **reflexive**
(again, in the logical sense of the term, not the grammatical one); if not,
then it is **irreflexive.** In principle, it is possible to build a
coherent formal system based on either answer. From the point of view of
syntactic theory, it is preferable to define dominance as reflexive because
it simplifies the definitions of linguistically relevant, derived relations
such as c-command and binding.

An important subcase of dominance is **immediate dominance.**
This is the case where the two nodes in question are connected by a single
branch without any intervening nodes. More formally, immediate dominance
is defined as in (2).

(2) | A immediately dominates B iff | ||

a. | A dominates B, and | ||

b. | there is no node C, distinct from A and B, such that A dominates C and C dominates B. |

Unlike dominance, immediate dominance is not a transitive relation.
This is apparent from even a simple structure like (1), where
`NP` immediately dominates `N`, and `N` immediately
dominates `Zelda`, but `NP` does not immediately dominate
`Zelda`.

In general, trees are more complex than the very simple case in (1), and they contain nodes that have more than one branch emanating from them, as in (3).

(3) |

In such trees, two nodes are related either by dominance or by a
second primitive relation, **precedence.** Precedence is
represented graphically in terms of left-to-right order. Dominance and
precedence are mutually exclusive. That is, if A dominates B, A cannot
precede B, and conversely, if A precedes B, A cannot dominate B. Like
dominance, precedence is a transitive relation, and just as with
dominance, there is a nontransitive subcase called **immediate
precedence.** The definition of immediate precedence is analogous to
that of immediate dominance; the term *dominates* in (2) is simply
replaced by *precedes.* The difference between precedence, which is
transitive, and immediate precedence, which isn't, can be illustrated in
connection with (3). The first instance of `Noun` (the one that
immediately dominates `secretary`) both precedes and immediately
precedes `TrVerb`, and `TrVerb` in turn both precedes and
immediately precedes the second instance of `NounPhr` (the one
that dominates `the letter`). The first instance of
`Noun` precedes the second instance of `NounPhr`, but not
immediately.

Certain relations among nodes are often expressed by using kinship
terms. If A dominates B, then A is the **ancestor** of B, and B is
the **descendant** of A. If A immediately dominates B, then A is
the **parent** of B, and B is the **child** of A. If A
immediately dominates B and C, then B and C are **siblings.**
Often, the female kinship terms **mother,** **daughter,** and
**sister** are used for the corresponding sex-neutral ones. In
(3), `Sentence` is the ancestor of every other node in the tree.
`Secretary` is the child of the first `Noun`. The first
`NounPhr` and `VerbPhr` are sisters, and so are
`TrVerb` and the second `NounPhr`, but `drafted`
and the second instance of `the` are not (they don't have the
same mother). Notice, incidentally, that syntactic trees are
single-parent families. Most theories of syntax do not allow nodes
with more than one parent.

Depending on the number of daughters, nodes are classified as either
**nonbranching** (one daughter) or **branching** (more than
one daughter). A more detailed system of terminology distinguishes
nodes that are **unary-branching** (one daughter),
**binary-branching** (two daughters), and
**ternary-branching** (three daughters). Nodes with more than
three daughters are hardly ever posited in syntactic theory. Indeed,
according to an influential hypothesis (Kayne 1984),
Universal Grammar allows at most binary-branching nodes. According to
this hypothesis, it is a formal
universal of human language that the number of branches associated
with any node cannot exceed 2.

Some node A **exhaustively dominates** two or more nodes *only* condition
(it dominates too much material). A also fails to exhaustively dominate B
C in (4d), because it runs afoul of the *all* condition (it dominates
too little material).

(4) | a. | b. | c. | d. |

As is evident from (4b,c), dominance is a necessary but not a sufficient condition for exhaustive dominance.

(5) | A c-commands B iff | ||

a. | neither A nor B dominates the other, and | ||

b. | the lowest branching node that dominates A also dominates B. |

Notice that the notion of c-command is defined in terms of dominance
and makes no mention of precedence. It is tempting to assume that
c-command logically implies precedence, or vice versa, but it is a
temptation to be firmly
resisted.^{2}<

Notice further that c-command is not necessarily a **symmetric**
relation. In other words, a node A can c-command a node B without B
c-commanding A. For instance, in (3), `VerbPhr` c-commands
`secretary` (because the first branching node dominating
`VerbPhr`, namely `Sentence`, dominates `secretary`),
but not vice versa (because the first branching node that dominates
`secretary`, namely the first instance of `NounPhr`, doesn't
dominate `VerbPhr`).

Although c-command isn't necessarily a symmetric relation, it is
possible for two nodes to c-command each other. This is the case when
the two nodes are sisters. Syntactic sisterhood is also known as
**mutual c-command** or **symmetric c-command.**

(6) | A binds B iff | ||

a. | A c-commands B, and | ||

b. | A and B are coindexed. |

The coindexing referred to in (6b) can arise either through coreference or through movement. These two cases are illustrated in (7).

(7) | a. | Lukas_{1} can comb himself_{1}.
| (coreference) | |

b. | Who_{i} did you see t_{i}?
| (movement) |

If A binds B, B is **bound** by A (not *bounded* !). If A
does not bind B, B is said to be **free.** B is **free in C** if
there is no A that binds B, with both A and B dominated by C.

(i) | A commands B iff | |||

i. | neither A nor B dominates the other, and | |||

ii. | the S(entence) node that most immediately dominates A also dominates B. |