| (1) |
|
The nodes in a tree like (1), where no node has more than one branch
emanating from it, 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 concept
like point in geometry. Dominance is represented graphically in terms of
top-to-bottom order. That is, if a node A dominates a node B, it 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). This means that 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 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 definition.
Which definition leads to a simpler system from a linguistic point of view
is a question that arises in connection with Problem 1.1, so we won't go into
the matter further here.
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 as simple a structure as (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 one in (1) and contain nodes that have more than one branch emanating from them, as is the case in (3).
| (3) | 
|
In such trees, two nodes can (and in fact must) be 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 transitive precedence and intransitive immediate precedence can be illustrated in connection with (3). Zelda both precedes and immediately precedes helped, and helped in turn both precedes and immediately precedes her. Zelda precedes her, 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 this book, we will use the two sets of terms interchangeably. In (3), S is the ancestor of every other node in the tree. Zelda is the child of the leftmost N. The leftmost NP and VP are sisters, and so are V and NP, but helped and her are not (they're first cousins once removed, if you will). Notice, incidentally, that syntactic trees are single-mother families (excepting the ancestorless root node). Nodes with more than one parent are not allowed in syntax, at least not ordinarily.
Depending on the number of daughters, nodes are classified as either nonbranching (one daughter) or branching (more than one daughter). An alternative, more detailed system of terminology that is also in use distinguishes nodes that are unary-branching (one daughter), binary-branching (two daughters), and ternary-branching (three daughters). Nodes with more than three daughters are very rarely used in theoretical syntax. Indeed, according to an influential hypothesis (Kayne 1984), Universal Grammar allows at most binary-branching nodes. That is, according to this hypothesis, binary-branchingness is a formal universal of human language.
A node A exhaustively dominates two or more nodes
| (4) | a. | 
| b. | 
| c. | 
| d. | 
|
As is evident, domination is a necessary but not a sufficient condition for exhaustive domination.