### Node relations

#### Dominance

It is convenient to represent syntactic structure by means of graphic structures called trees; these consist of a set of nodes connected by branches. It is sometimes useful to distinguish between two types of nodes: terminal nodes, which are labeled with vocabulary items, and nonterminal nodes, which are labeled with syntactic categories. In a very simple tree like (1), the only terminal node is labeled Zelda, and the two nonterminals are labeled N and NP.

 (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.

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 (= if and only if) 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.

#### Precedence

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.

#### Kinship terminology

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.

#### Branching

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.

#### Exhaustive dominance

Some node A exhaustively dominates two or more nodes B, C, ... iff (= if and only if) A dominates all and only B, C, ... For instance, A dominates the string B C in (4a-c), but exhaustively dominates it only in (4a). A doesn't exhaustively dominate B C in (4b,c), because it runs afoul of the 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.

#### C-command

A derived relation that is central to syntactic theory is c-command,1 which is defined as follows.

 (5) A c-commands B iff (= if and only if) 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 (= if and only if) 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.

#### Notes

1. The odd name c-command is short for 'constituent-command' and reflects the fact that the c-command relation is a generalization of a relation (now obsolete) called command, defined as in (i) (Langacker 1969:167).

 (i) A commands B iff (= if and only if) i. neither A nor B dominates the other, and ii. the S(entence) node that most immediately dominates A also dominates B.
2. Recently, an axiom has been proposed - the so-called Linear Correspondence Axiom - according to which c-command relations be expressed as precedence relations (Kayne 1994). Contrary to what is sometimes believed, this proposal underscores the logical independence of c-command and precedence rather than eliminating it. If precedence were logically derivable from c-command, there would be no need for an axiom postulating a correspondence between them (indeed, given the meaning of the term 'axiom', such an axiom would be a contradiction in terms).