Structural Relations
The mathematical properties of
phrase structure trees
Important!!
This is the most mathematical and
technical topic we will cover.
 Even if you have trouble with the formal
definitions. Try to understand the
INTUITIVE idea behind them. Don’t get
lost in the details of the formalism.

Structural Relations
Structural relations: the formal
relationships between items of a tree
 Why should we care? We want to be
able to talk about specific relationships
in terms of structures.
 Structural relationships are actually very
simple! Don’t let the formalism scare
you!

Some basic terms
M
Root node
Branches
Non-terminal nodes
N
D
E
O
F
Labels: M,N,O,D,E,F,G,H,J
Node: Any point with a label
G
H
J
Terminal nodes
Dominance

Intuitively: this is containment. If a node
contains another, then it dominates it:
A
A dominates
B,C,D,E,F,G
D dominates
E,F,G
[A B C [D E F G]]
contained
inside [A ]
B
C
D
E
F
G
Dominance

Other intuitive definitions:
» “On top of”,
» “Can follow a branch downwards to”.
Dominance

A slightly more formal definition:
» Dominance: Node A dominates node
B if and only if A is higher up in the
tree than B and if you can trace a line
from A to B going only downwards.
Immediate Dominance
»
Node A immediately dominates node B if
there is no intervening node G which is
dominated by A, but dominates B
»
(in other words, A is the first node that dominates B)
A
B
C
D
A dominates B,C,D,E,F,G
but A immediately dominates only
B,C,D
E
F
G
Properties of domination




x ≤ x (every linguistic element dominates itself)
if x ≤ y ≤ z then x ≤ z (the dominance relation is transitive)
if x ≤ y ≤ x then, x = y (the dominance relation is unidirectional)
if x ≤ z and y ≤ z then either x ≤ y or y ≤ x (or both if x =y =z)
(multiple mothers are ruled out)
A
* A
B
B
C
C
Exhaustive Domination

Node A exhaustively dominates a SET of
nodes {B,C,…,D},
provided it immediately dominate all the
members of the set (so that there is no
member of the set that is not dominated by A)
» AND there is no node G immediately
dominated by A that is not a member of the
set.
»
Exhaustive Domination
F
A
B
C
G
D
E
H
I
A exhaustively dominates the set {B,C,D,E}
A does NOT exhaustively dominate the set {B,C,D}
A does NOT exhaustively dominate the set {B,C,D, E, H}
A formal definition of
Constituency

Constituent: A set of nodes exhaustively
dominated by a single node.
F
A
B
C
{E, H} are NOT
a constituent
G
D
E
H
I
Constituent vs Constituent of
Constituent of does NOT mean the
same thing as constituent.
 Essentially constituent of is the opposite
of domination.
 A dominates B, then we say B is a
constituent of A.
 immediate constituent of is the opposite
of immediate domination.

Some Informal Terms
Mother: the node that immediately
dominates another.
 Daughter: the node that is immediately
dominated by another (is an immediate
constituent of another).
 Sisters: two nodes that share the same
mother.

Root and Terminal Nodes
Root node: A node with no mother
 Terminal node: A node with no
daughters

Root Node
S
NP
VP
D
N
the platypus
V
laughed
Terminal Nodes
Precedence

Precedence: Node A precedes node B if and
only if
The intuitive part
» (1) A is to the left of B
» (2) and neither A dominates B nor B
dominates A
» (3) and every node dominating A either
precedes B or dominates B
why do we need clauses (2) and (3)?
Breaking down the definition
of precedence

(2) and neither A dominates B nor B
dominates A
Is the ball to the left or right of the box?
Neither! You can’t precede or follow something that
dominates (contains) you or you dominate (contain).
Breaking down the definition
of precedence

(3) and every node dominating A either
precedes B or dominates B.
S
NP
D
the
VP
N
clown
V
kissed
Does kiss precede clown?
Obviously not!
NP
D
N
the donkey
“and every node dominating
kissed either precedes clown
or dominates clown”
Not true
No Crossing Branches
Constraint

If one node X precedes another node Y then
X and all nodes dominated by X must
precede Y and all nodes dominated by Y.
*
M
N
P
O
R
Q
S
Properties of precedence




if xy, then NOT(yx) (You can’t both precede and follow)
if x y z, then x z (precedence is transitive)
if xy or y x, then NOT(x≤y) and NOT (y≤x) (precedence and
dominance are mutually exclusive) there is a mistake in the textbook on this one.
x y iff for all terminals u, v, x ≤ u and y ≤ v jointly imply u v (don’t
cross lines)
A
B
C
Immediate Precedence

Immediate Precedence:
» A immediately precedes B if there is no
node G which follows A but precedes B.
A
B
G
A
G
B
C-command

Intuitively: The relationship between a
node and its sister, and all the daughters
of its sister
A c-commands C,D,E,F,G,H
M
A
Note: D does NOT
c–command A
C
D
E
F
G
H
C-command

Node A c-commands node B if every
branching node dominating A also
dominates B, and A does not itself
dominate B.
you can’t command something you dominate
Sisterhood
Symmetric C-command
A symmetrically c-commands B, if A ccommands B AND B c-commands A
 SAME THING AS SISTERHOOD

M
A
A & B symmetrically c-command one another
B
D
A does NOT symmetrically c-command D
E
F
G
H
Asymmetric C-command

A asymmetrically c-commands B, if A ccommands B but B does NOT c-command
A.

(intuitively -- A is B’s aunt)
M
A
C
B
E
F
G
H
Grammatical Relations
Subject: NP daughter of S
 Object: NP daughter of VP
 Object of a Preposition: NP daughter of
PP

Summary
Grammatical Relations: relationship
between nodes.
 Dominance (=containment)

immediate dominance (=motherhood)
» exhaustive dominance (=constituent)
»

Precedence (=to the left)
»
immediate precedence (=adjacent & to the
left)
Summary

C-command: Sisters & nieces
Symmetric C-command: sisters
» Asymmetric C-command: Aunt asymm. ccommands nieces
»

Grammatical Relations
Subject: NP daughter of S
» Object: NP daughter of VP
» Object of P: NP daughter of PP
»