How to find the number of trees of a graph?

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What is the process of finding the number of trees of a graph?

Q: Amongst all trees with 10 vertices {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} consider those which have a vertex of degree 8. What is the number of such trees? Justify your answer.

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Start by drawing a vertex of degree 8:

This uses 9 of the 10 vertices. You need to add a tenth vertex in such a way that you still have a tree and you still have a vertex of degree 8. You also want to add labels to it.