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A Chomsky-normal-form parse tree of height $h$ has yield length at most $2^{h - 1}$

Statement

Let $G$ be a context-free grammar in Chomsky normal form, and let $T$ be a parse tree for $G$ of height $h \geq 1$ . If $y (T)$ is the yield of $T$ , then $∣ y (T) ∣ \leq 2^{h - 1} .$

Facts & Assumptions

Given: A Chomsky-normal-form grammar $G$ and a parse tree $T$ of height $h \geq 1$ .

[L1]

By Parse trees and their yields, the yield of a parse tree is the left-to-right list of its terminal leaves, and every internal node is labeled by a variable together with one production used at that node.

[L2]

By Chomsky normal form, every production in a CNF grammar is either $A \to B C$ , $A \to a$ , or the exceptional start rule $S \to ε$ .

Proof

technique · direct

1.1

We induct on the height $h$ . If $h = 1$ , then the root is already a leaf. By [L1], a leaf at the root can only be the variable leaf corresponding to the exceptional CNF rule $S \to ε$ from [L2]. Thus $y (T) = ε$ , so $∣ y (T) ∣ = 0 \leq 1 = 2^{0}$ .

L1L2given

2.1

Suppose $h > 1$ . If the root uses a terminal production $A \to a$ , then the yield has length $1$ , so the bound is immediate. Otherwise the root uses a binary production $A \to B C$ by [L2]. The two child subtrees have heights at most $h - 1$ , so by the induction hypothesis each contributes at most $2^{h - 2}$ terminal leaves. Therefore $∣ y (T) ∣ \leq 2^{h - 2} + 2^{h - 2} = 2^{h - 1}$ .

L1L2step 1.1induction

3.1

The bound holds for height $1$ and is preserved when the height increases, so every CNF parse tree of height $h$ has yield length at most $2^{h - 1}$ .

step 1.1step 2.1∎

Depends on

Used by

The pumping lemma for context-free languages Theorem

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jean Gallier and Jocelyn Quaintance, Introduction to the Theory of Computation: Some Notes for CIS511 (standard reference, not scraped)
Alfred V. Aho, COMS W3261 CS Theory, Lecture 10 (standard reference, not scraped)

A Chomsky-normal-form parse tree of height hh has yield length at most 2h−12h−1