There are several results on the structure of graphs that exclude a tree or forest as a minor. However, there is little data on what graphs excluding specific small trees look like. Information of this kind can help to generate new general questions. In Graph Minors I, Robertson and Seymour proved that every graph that does not have a forest as a minor has bounded path-width. In 2023, Dujmović et. al. strengthened the bound.

In this talk, we give a specific example of excluding a particular tree as a subgraph. We also raise questions about path-width of classes of graphs that exclude a particular tree.