Consider the grammar. S -> SxSxS | ϵ For a string “xxxxxx”, How many total…
Consider the grammar.
S -> SxSxS | ϵ
For a string “xxxxxx”, How many total number of parse tree/s are possible.
Answer: 12 — Key insight: each application of the rule S -> S x S x S adds two terminal symbols 'x'. Step 1: Determine how many times the rule must be applied. Let k be…
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Correct answer: 12
Key insight: each application of the rule S -> S x S x S adds two terminal symbols 'x'.
Step 1: Determine how many times the rule must be applied.
Let k be the number of times S -> S x S x S is used. Each use contributes 2 x's, so 2k = 6, giving k = 3.
Step 2: Count the number of distinct parse trees for k = 3.
Each internal node of the parse tree has exactly three ordered children (a full ordered ternary tree). The number of such trees with k internal nodes is given by the Fuss–Catalan (generalized Catalan) formula:
Number = 1/((3-1)k + 1) * binomial(3k, k) = 1/(2k + 1) * C(3k, k).
For k = 3: binomial(9, 3) = 84, and 2k + 1 = 7, so the number = 84 / 7 = 12.
Conclusion: There are 12 distinct parse trees that generate the string "xxxxxx".