In this chapter we study spaces that are related to rooted trees. These include the rooted trees themselves, one of the simplest types of graphs, as well as the corresponding partial order graphs. In addition, we study downward run graphs and upward run graphs constructed from rooted trees. These graphs generalize the remaining-life and success-run graphs in classical reliability theory. We also consider free semigroups, one of the simplest types of semigroups since the associated graph is a rooted tree.