Definition: transient random walk What exactly does a "transient random walk on a graph/binary tree" mean? Does it mean that we never return to the origin (assuming there is one as for the tree) or just any vertex of the graph or tree? Thanks.

You can define transience on an infinite graph, but you need sufficient conditions on the graph so that transience means the same thing for all vertices. For example, if you can go from every vertex to every vertex with positive probability, then this is sufficient.
(In particular, if you have a simple random walk on the graph, it is sufficient to suppose that the graph is connected and locally finite, that is, any two vertices are joined by a finite path and all degrees are finite.)
On a finite connected graph the random walk is always recurrent.
