> any two vertices are connected by (i.e. are the endpoints of) exactly one path No. You just described a connected acyclic graph, not a tree. In addition to

> any two vertices are connected by (i.e. are the endpoints of) exactly one path No. You just described a connected acyclic graph, not a tree. In addition to being connected and acyclic, a tree must also have a root, and is thus implicitly directed.