In Hatcher's - a maximal sub-tree in a graph is defined as a contractible subgraph that reaches all the vertices. Later he shows that this definition is equivalent to a cycle-free connected graph - using the idea that a tree may be constructed by taking a root and adding "layers" (the neighobours of the root, and then their neighbours and so on) - as in the proof of existense of a maximal tree. But when wanting to prove that a tree is cycle-free, we are given an arbitrary tree - and it sort of seems to me like he assumes "rooted-tree" structure on it. If we knew that a maximal tree is a tree that is not contained in any other tree, i can see how he can show the existense of this rooted tree structure - but i dont know how to show this statement without using the cycle-free definition.
I'd appriciate any clearance on this