Is there any universally agreed definition of "length" (or "width", or whatever term) of an element in a free group $F_n(x_1,\cdots,x_n)$? Intuitively, I would like the length of $1$ to be $0$; the length of $x_2^{-3}x_1^4x_2x_1^2x_2$ to be $5$ and the length of $x_1^{-4}x_2x_1^2x_2$ to be $4$, etc.
Basically the idea is that the "length" of an element $W$ is the sum of the occurrences of each $x_i$ in the reduced word which is equivalent of $W$. Is there any clearer definition?
This is the word metric, though with respect to $S=\{\,x_i^k\mid 1\le i\le n, k\in\mathbb N\,\}$ instead of the generators only.