Counting Spanning Trees Kun-Mao Chao ( 趙坤茂 ) Department of Computer Science and Information Engineering National Taiwan University, Taiwan WWW:

2 Spanning Trees A spanning tree for a graph G is a subgraph of G that is a tree and contains all the vertices of G.

3 All 16 spanning trees of K 4

4 Cayley’s formula Back in 1889, Cayley devised the well- known formula n n-2 for the number of spanning trees in the complete graph K n The first explicit combinatorial proof of Cayley's formula was given by Pr\"{u}fer in 1918.

5 Pr\"{u}fer sequence A Pr\"{u}fer sequence of length n-2, for n >= 2, is any sequence of integers between 1 and n, with repetitions allowed. There are n n-2 Pr\"{u}fer sequences of length n-2.

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10 Try the following Pr\"{u}fer sequences Assume that the vertex set is {1, 2,3, 4, 5, 6, 7, 8} (a line) (two-star) (star)

11 Knuth’s talk on counting spanning trees Donald E. Knuth gave a lecture on counting spanning trees in December (In fact, he likes to talk about trees in the Christmas season. Christmas trees …)Donald E. Knuth Try it?