Is there any special word for a maximal non-branching directed path in a network or diforest?

To be 100% precise, by "maximal non-branching directed path" I mean a path $P=x_1,x_2,\ldots$ (maybe finite, maybe infinite) such that

- Every vertex in the path (except possibly the terminal vertex of the path (if there is one)) has outdegree 1.
- The path is directed (for every $x_i,x_{i+1}$ in the path, there is an edge directed from $x_i$ to $X_{i+1}$).
- There is no $x_0$ such that $P'=x_0,x_1,\ldots$ has properties 1--2.
- If $P$ is finite (say $P=x_1,\ldots,x_n$) then there is no $x_{n+1}$ such that $x_1,\ldots,x_{n+1}$ has properties 1--2.

If I were king of the world I would decree that such things are "segments", but that terminology doesn't seem to exist in the literature.