I need to find the number of spanning trees on $V = \{1,2,3,4,5,6,7,8,9\}$, where $\{1,2,3,4\}$ are leaves. Can anyone tell me how?
2026-03-27 07:18:42.1774595922
Finding the number of spanning trees on a set of vertices.
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Work backwards. Erase the vertices $1,2,3,4$. Then, since all four are leaves, you get a spanning tree on $\{5,6,7,8,9\}$.
This spanning tree is a spanning tree of $K_5$, and the number of such spanning trees is well known. Now, add back the vertices $1,2,3,4$. Since they are leaves, each of them can be connected to one of $\{5,6,7,8,9\}$, and the choices are independent. Thus, for each of them you have exactly $5$ choices.