Final answer:
There are two nonisomorphic unrooted trees with three vertices: a linear tree and a star tree, which are distinguishable due to the different degrees of their vertices.
Step-by-step explanation:
The question asks how many nonisomorphic unrooted trees there are with three vertices. This is a combinatorial mathematics problem that involves understanding the concept of isomorphic graphs and trees.
In graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path. For a tree with three vertices, there are only two distinct shapes it can take:
Since these two trees are not identical in shape and cannot be transformed into one another by simply renaming the vertices (i.e., they are not isomorphic), we conclude that there are two nonisomorphic unrooted trees with three vertices. The trees' lack of rootedness means that there is no designated root vertex that would change the way we count different shapes.
The ability to distinguish between these two forms relates to the trees' degrees of vertices. In a linear tree, the two end vertices have a degree of 1, and the middle vertex has a degree of 2, whereas in a star-shaped tree, the middle vertex has a degree of 2, and the two outer vertices each have a degree of 1. Since vertices with different degrees cannot be equivalent in an isomorphism, the trees remain nonisomorphic.