First we should consider if this even makes sense. Given any connected graph \(G\text{,}\) will there always be a subgraph that is a tree? Well, that is actually too easy: you could just take a single vertex of \(G\text{.}\) If we want to use this subgraph to tell us how to visit all vertices, then we want our subgraph to include all of the vertices. We call such a tree a spanning tree. It turns out that every connected graph has one (and usually many).
in-context