Chapter 3 Context-Free Grammars, Context-Free Languages, Parse ...

66 CHAPTER 3. CONTEXT-FREE GRAMMARS AND LANGUAGES
is shown in Figure 3.1. The yield of the derivation tree is a + a ∗ a.
E
T
F
T
F
E
+
a a
Figure 3.1: A complete derivation tree
Derivations trees are associated to derivations inductively as follows.
Definition 3.11.2 Given a context-free grammar G =(V,Σ,P,S), for any A ∈ N, ifπ :
A n
=⇒ α is a derivation in G, we construct an A-derivation tree tπ with yield α as follows.
(1) If n =0,thentπ is the one-node tree such that dom(tπ) ={ɛ} and tπ(ɛ) =A.
(2) If A n−1
=⇒ λBρ =⇒ λγρ = α, thenift1 is the A-derivation tree with yield λBρ associated
with the derivation A n−1
=⇒ λBρ, andift2is the tree associated with the production
B → γ (that is, if
γ = X1 ···Xk,
then dom(t2) ={ɛ, 1,...,k}, t2(ɛ) =B, andt2(i) =Xi for all i, 1≤ i ≤ k, orifγ = ɛ,
then dom(t2) ={ɛ, 1}, t2(ɛ) =B, andt2(1) = ɛ), then
∗
T
a
F
tπ = t1[u ← t2],
where u is the address of the leaf labeled B in t1.
The tree tπ is the A-derivation tree associated with the derivation A n
=⇒ α.
Given the grammar
where P is the set of rules
G2 =({E,+, ∗, (, ),a}, {+, ∗, (, ),a},P,E),
E −→ E + E,
E −→ E ∗ E,
E −→ (E),
E −→ a,