Give an example to show that the second part of Lemma 84.2 need not hold if \(T\) is infinite.
Solution
Let \(T\) be the bi-infinite linear graph with vertex set \(\mathbb Z\) and one edge joining \(n\) to \(n+1\) for each integer \(n\). This graph is connected and contains no closed reduced edge path, so it is a tree.
Suppose the converse conclusion of Lemma 84.2 held: suppose \[ T=T_0\cup A, \] where \(T_0\) is a tree and the edge \(A\) meets \(T_0\) in exactly one vertex. Let the other endpoint of \(A\) be \(v\). Since \(v\notin T_0\), every edge of \(T\) incident to \(v\) must equal \(A\). Thus \(v\) would have degree \(1\) in \(T\).
But every vertex of the bi-infinite line has degree \(2\). This contradiction shows that no such decomposition exists. Hence the finiteness hypothesis in the second part of Lemma 84.2 is essential.