There are uncountably many topological types of locally finite trees

Consider two locally finite rooted trees as equivalent if each of them is a topological minor of the other, with an embedding preserving the tree-order. Answering a question of van der Holst, we prove that there are uncountably many equivalence classes. Key words: graph, locally finite tree, embedding, wqo, bqo Let the tree-order on the set of vertices of a rooted tree T be defined by setting x y for vertices x, y iff x lies on the unique path in T from its root to y. Let us call two locally finite rooted trees equivalent if each of them is a topological minor of the other, with an embedding that respects the treeorder. Call the equivalence classes topological types of such trees. The purpose of this note is to answer a question raised by van der Holst [2], by proving that there are uncountably many topological types of locally finite trees. Our proof uses Nash-Williams's theorem that the

• Type: Journal
• Year: 2006
• Where: JCT
• Authors: Lilian Matthiesen
• Keywords: Finite Rooted Trees, JCT 2006, Rooted Trees, Van Der Holst