(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.
Dominance is a transitive relation (in the logical sense of
the term, not the grammatical one). In other words, if A dominates B,
and if B dominates C, then it is necessarily the case that A dominates
C.
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
(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.
Binding
An important derived relation that is defined in terms of c-command is the
notion of binding.
(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. | Lukas1 can comb himself1. | (coreference) | |
b. | Whoi did you see ti? | (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. |