Suppose that $G$ is a context-free grammar without $\epsilon$-productions. Given a derivation of some word $w$ in the language $L$ of $G$, describe a parse tree for this derivation.

State and prove the pumping lemma for $L$. How would your proof differ if you did not assume that $G$ was in Chomsky normal form, but merely that $G$ has no $\epsilon$ - or unit productions?

For the alphabet $\Sigma=\{a, b\}$ of terminal symbols, state whether the following languages over $\Sigma$ are context free, giving reasons for your answer. (i) $\left\{a^{i} b^{i} a^{i} \mid i \geqslant 0\right\}$, (ii) $\left\{a^{i} b^{j} \mid i \geqslant j \geqslant 0\right\}$, (iii) $\left\{w a b w \mid w \in\{a, b\}^{*}\right\}$.