Maximizing Emergency Response Efficiency in Urban Planning

1. Graph Theory Chapter 3 Trees and Forests 大葉大學 資訊工程系 黃鈴玲 2011.9

2. Contents • 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

3. 3.1 Trees and Some of Their Basic Properties 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. 3.2 Characterizations of Trees Theorem 3.7 Proof

9. 3.3 Inductive Proofs on Trees Example 3.8 Regular binary tree: all vertices have degree 3 or less. Let d3(n) denote the maximum number of vertices of degree 3 that such a tree T on n vertices can have. Then Proof (see Ex3.5) Let x, y, and z be the number of vertices in T of degree 1, 2, 3. Then x+y+z=nandx+2y+3z=2n-2, z n/2- 1 y+2z=n-2 2z n-2

10. 3.5 Centers in Trees

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-pathin G.

12. G: Question A model of a street system: edge: streetvertex: intersection Q: How to place the police station and fire station?

13. How to choose the locations? 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 minimizemax{d(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e(u)減少1

15. Theorem 3.20 Theorem 3.21

16. ExerciseFind the distance of u,v, and their eccentricities. u v

17. Exercise Find all centers of the graph. Exercise

18. Definition (直徑及半徑) The diameter of a graph G is diam(G) = max{ d(u, v) : u, v  V(G) } = max{e(u) : u  V(G) } The radius of a graph G is rad(G) = min{e(u) : u  V(G) } Exercise Find the diameters and radii of the graphs in Exercise 3.21 and 3.22.

19. 3.6 Rooted Trees Definition 3.22 Example 3.23

20. Definition 3.24

21. Example 3.25

22. Definition 3.26 Example 3.27

23. 3.7 Binary Trees Definition 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 Trees Definition 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) (對照下一頁的圖) (b) Every vertex except the leaves has exactly two children, each level (except level 0) must contain at least two vertices.  ht(T) (n-1)/2  n2ht(T) + 1

28. Example 3.39

29. Definition 3.41 (level 0~k-1都全滿, level k 不要求) Observation