Counting Spanning Trees

(1)

Counting Spanning Trees

Kun-Mao Chao ( 趙坤茂 )

Department of Computer Science an d Information Engineering

National Taiwan University, Taiwan

E-mail: [email protected]

(2)

2

Spanning Trees

• A spanning tree for a graph G is a subgraph of G that is a tree and contains all the vertic es of G.

(3)

3

(4)

4

Cayley’s formula

• Back in 1889, Cayley devised the well-kno

wn formula nn-2 for the number of spanning

trees in the complete graph K_n

• The first explicit combinatorial proof of Ca yley's formula is due to Pr\"{u}fer.

(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 nn-2 Pr\"{u}fer sequences of lengt

(6)

(7)

(8)

(9)

(10)

10

Try the following Pr\"{u}fer

sequences

Assume 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 spanning}

trees last December. (In fact, he likes to talk about trees in the Christmas season.

Christmas trees …)

(12)

12

元宵燈謎 ( 字謎 )

• 路上行人走，小月照其上 • 兩人共有十五顆心 • 日落半林中　 • 太陽掛在樹頂上 • 李字少了木，不作子字猜 • 百年前是草，百年後是木

(13)

13

元宵燈謎 ( 字謎 )

• 路上行人走，小月照其上　　　　　趙 • 兩人共有十五顆心　　　　　　　　德 • 日落半林中　　　　　　　東 • 太陽掛在樹頂上　　　　　　　　　果 • 李字少了木，不作子字猜　　　　　一 • 百年前是草，百年後是木　　　　　葉

(14)

14

回應與挑戰（一）

同學：停電，猜一字趙老：．．．同學：請用英文想趙老：．．．同學：停電是 black out 趙老：原來是黑出，黜！

(15)

15

回應與挑戰（二）

趙老：我在水中，猜一字同學：．．．趙老：請用英文想同學：．．．趙老：是一個英文字， I in the WATER 同學：原來是 WAITER ！