Term for the tree of all paths starting from a given vertex in a graph
It's the self-avoiding walk tree (SAW tree).

The trees seem to have been first considered by C.D.&nbsp;Godsil (Matchings and Walks in Graphs, Journal of Graph Theory 5(3):285&ndash;297, 1981; DOI link; just before Lemma&nbsp;2.4), though Godsil doesn't use the name.

(And, yes, "self-avoiding walk" is just a mighty-long way of saying "path".)