1. Counting Spanning Trees Kun-Mao Chao ( 趙坤茂 ) Department of Computer Science and Information Engineering National Taiwan University, Taiwan E-mail: kmchao@csie.ntu.edu.tw WWW: http://www.csie.ntu.edu.tw/~kmchao

2. 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. 3 All 16 spanning trees of K 4

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. 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.

6. 6

7. 7

8. 8

9. 9

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

11. 11 Knuth’s talk on counting spanning trees Donald E. Knuth gave a lecture on counting spanning trees in December 2003. (In fact, he likes to talk about trees in the Christmas season. Christmas trees …)Donald E. Knuth Try it?