In graph theory, a caterpillar or caterpillar tree is a tree in which all the vertices are within distance 1 of a central path. Wikipedia

A mathematical tree in which all leaves are connected to a central path Wikipedia Disambiguation

A tree consisting of only a path (the spine or stalk of the tree) and vertices directly connected to (i.e. one edge away from) that path; a tree whose vertices with a degree of at least 3 are all surrounded by at most two vertices of degree two or greater. Wiktionary