The root has to come at the end of one of the branches. So there are nine possible choices of root (see the attachment). That gives you nine possible rooted trees. But some of them are isomorphic. For example, the tree with vertex 2 as its root is isomorphic to the tree with vertex 5 as its root, because you can get from one to the other by twisting those two branches so that they swap positions. Similarly, the trees with roots at 6, 7 and 8 are all isomorphic. Are there any other isomorphisms among these trees?
I do not know how to attach the tree but if you inbox me can send it to you there.
Jul 29th, 2014
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