A string is repeat$-$free if non$-$empty prefixes of the string does not occur in the rest of the string. That is$,$ w is repeat$-$free if for all x$,$ y$,$ if xy $=$ w then x does not occur in y$.$

How many prefix$-$free strings of length n are there for each of the following alphabets? Show your work.

$$ The binary alphabet $\{0,1\}$

$$ Theternaryalphabet$\{0,1,2\}$