Dynamic programming

Fill out the table described in the polynomial time algorithm for context-free language recognition from Theorem 7.16 for string $w=$ baba and CFG $G$ :

Show that P is closed under the star operation. (Hint: Use dynamic programming. On input $y=y_{1} \cdots y_{n}$ for $y_{i} \in \Sigma$, build a table indicating for each $i \leq j$ whether the substring $y_{i} \cdots y_{j} \in A^{*}$ for any $A \in \mathrm{P}$.)

Let UNARY-SSUM be the subset sum problem in which all numbers are represented in unary. Why does the NP-completeness proof for SUBSET-SUM fail to show UNARY-SSUM is NP-complete? Show that UNARY-SSUM $\in \mathrm{P}$.

Modify the algorithm for context-free language recognition in the proof of Theorem 7.16 to give a polynomial time algorithm that produces a parse tree for a string, given the string and a CFG, if that grammar generates the string.