CFG & PDA formulas

For $|w|=n\ge 1$ and any derivation of $w$ from $S$ in a CNF grammar: $n-1$ steps use $A\to BC$ and $n$ steps use $A\to a$. In the parse tree there are $n-1$ variable nodes with two variable children and $n$ variable nodes whose only child is a terminal leaf, so $2n-1$ variable nodes, one per step.

For $|w|=n\ge 1$; each step adds exactly one terminal.

Every CFG has an equivalent NPDA and every NPDA has an equivalent CFG, with either acceptance mode.