Recursive definition for the length of a string?
Strings are concatenations of characters. Hence, any non-empty string $w$ has a first character $x$ and can be written as $w = x w'$, where $w'$ is the tail string. It follows from the definition of a string's length that $|w| = |x w'| = |x| + |w'| = 1 + |w'|$.