Chapter 3 Trees and Forests

大葉大學 資訊工程系 黃鈴玲2011.9

3.1 Trees and Some of Their Basic Properties 3.2 Characterizations of Trees 3.3 Inductive Proofs on Trees 3.5 Centers in Trees 3.6 Rooted Trees 3.7 Binary Trees 3.8 Levels in Rooted and Binary Trees

2

3

Definition 3.1

4

star

Ex 3.3

5

Example 3.2字典找字的方式： a rooted tree

6

Definition 3.3

Theorem 3.4

Ex 3.18

7

Lemma 3.6

8

Theorem 3.7

Proof

9

Example 3.8

Proof

Regular binary tree: all vertices have degree 3 or less.

Let d3(n) denote the maximum number of vertices of degree 3that such a tree T on n vertices can have. Then

12

)(3

n

nd

(see Ex3.5)

Let x, y, and z be the number of vertices in T of degree 1, 2, 3.

Then x+y+z=n and x+2y+3z=2n2,

y+2z=n2 2z n2 z n/2 1

10

11

Definition 3.15

G: a graph. For u, v V(G), the distance between u and v,denoted (u,v), is the length of the shortest u, v-path in G.

G:

A model of a street system:

Q: How to place the police station and fire station?

edge: streetvertex: intersection

Minimize the response time between thefacility and the location of a possible emergency( 以出發後能最快到達事故地點為訴求 )(choose x to minimize max{(x,v) | v V(G) })

14

Definition 3.18 ( 離心率及中心 )

Example 3.19 Tree 中 eccentricity 值最大的一定發生在 leaves

removing all leaves, 使 (u) 減少 1

15

Theorem 3.20

Theorem 3.21

16

Exercise Find the distance of u,v, and their eccentricities.

u

v

17

Exercise

Exercise

Find all centers of the graph.

18

Definition ( 直徑及半徑 )

The diameter of a graph G is

diam(G) = max{ (u, v) : u, v V(G) } = max{(u) : u V(G) }

The radius of a graph G is

rad(G) = min{(u) : u V(G) }

Exercise Find the diameters and radii of the graphs in Exercise 3.21 and 3.22.

19

3.6 Rooted Trees3.6 Rooted TreesDefinition 3.22

Example 3.23

20

Definition 3.24

21

Example 3.25

22

Definition 3.26

Example 3.27

23

3.7 Binary Trees3.7 Binary TreesDefinition 3.28

Figure 3.15

24

Definition 3.29

Ex 3.33 Draw the regular binary trees on nine vertices.

25

Theorem 3.31

Pf.

若 tree 有 k 個 internal vertex ，則 每個 internal vertex 有 2 個 children ， 故 tree 共有 2k 個點是 children ( 因每個 child 只有一個 parent ，所以只會被計算一次 ) root 沒有 parent ，還沒被計算到

∴tree 共有 2k +1 個點 共有 k +1 個 leaves

26

3.8 Levels in Rooted and Binary 3.8 Levels in Rooted and Binary TreesTreesDefinition 3.37

27

Observation 3.38

Pf.

(a) If ht(T)= k, then n 20+21+…+2k = 2k+1 1

∴ n+1 2k+1 lg(n+1) k+1 lg(n+1)1 ht(T) (b) Every vertex except the leaves has exactly two children, each level (except level 0) must contain at least two vertices.

n 2ht(T) 1 ht(T) (n1)/2

( 對照下一頁的圖 )

28

Example 3.39

29

Definition 3.41

Observation

(level 0 ~ k1 都全滿 , level k 不要求 )