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1 Discrete Mathematics Trees H. Turgut Uyar Ayşegül Gençata Yayımlı Emre Harmancı
2 License You are free to: c T. Uyar, A. Yayımlı, E. Harmancı Share copy and redistribute the material in any medium or format Adapt remix, transform, and build upon the material Under the following terms: Attribution You must give appropriate credit, provide a link to the license, and indicate if changes were made. NonCommercial You may not use the material for commercial purposes. ShareAlike If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original. For more information: Read the full license:
3 Topics 1 Trees Introduction Rooted Trees Binary Trees Decision Trees 2 Tree Problems Minimum Spanning Tree
4 Topics 1 Trees Introduction Rooted Trees Binary Trees Decision Trees 2 Tree Problems Minimum Spanning Tree
5 Tree Definition tree: connected graph with no cycle examples
6 Tree Theorems Theorem T is a tree (T is connected and contains no cycle). There is one and only one path between any two distinct nodes in T. T is connected, but if any edge is removed it will no longer be connected. T contains no cycle, but if an edge is added between any pair of nodes one and only one cycle will be formed.
7 Tree Theorems Theorem E = V 1 proof method: induction on the number of edges
8 Tree Theorems Proof: Base step. E = 0 V = 1 E = 1 V = 2 E = 2 V = 3 assume that E = V 1 for E k
9 Tree Theorems Proof: Base step. E = 0 V = 1 E = 1 V = 2 E = 2 V = 3 assume that E = V 1 for E k
10 Tree Theorems Proof: Induction step. E = k + 1 remove edge (y, z): T 1 = (V 1, E 1 ), T 2 = (V 2, E 2 ) V = V 1 + V 2 = E E = ( E 1 + E 2 + 1) + 1 = E + 1
11 Tree Theorems Proof: Induction step. E = k + 1 remove edge (y, z): T 1 = (V 1, E 1 ), T 2 = (V 2, E 2 ) V = V 1 + V 2 = E E = ( E 1 + E 2 + 1) + 1 = E + 1
12 Tree Theorems Proof: Induction step. E = k + 1 remove edge (y, z): T 1 = (V 1, E 1 ), T 2 = (V 2, E 2 ) V = V 1 + V 2 = E E = ( E 1 + E 2 + 1) + 1 = E + 1
13 Tree Theorems Proof: Induction step. E = k + 1 remove edge (y, z): T 1 = (V 1, E 1 ), T 2 = (V 2, E 2 ) V = V 1 + V 2 = E E = ( E 1 + E 2 + 1) + 1 = E + 1
14 Tree Theorems Proof: Induction step. E = k + 1 remove edge (y, z): T 1 = (V 1, E 1 ), T 2 = (V 2, E 2 ) V = V 1 + V 2 = E E = ( E 1 + E 2 + 1) + 1 = E + 1
15 Tree Theorems Proof: Induction step. E = k + 1 remove edge (y, z): T 1 = (V 1, E 1 ), T 2 = (V 2, E 2 ) V = V 1 + V 2 = E E = ( E 1 + E 2 + 1) + 1 = E + 1
16 Tree Theorems Theorem T is a tree (T is connected and contains no cycle). T is connected and E = V 1. T contains no cycle and E = V 1.
17 Tree Theorems Theorem In a tree, there are at least two nodes with degree 1. Proof. 2 E = v V d v assume: only 1 node with degree 1: 2 E 2( V 1) E 2 V 1 E V 1 2 > V 1
18 Tree Theorems Theorem In a tree, there are at least two nodes with degree 1. Proof. 2 E = v V d v assume: only 1 node with degree 1: 2 E 2( V 1) E 2 V 1 E V 1 2 > V 1
19 Tree Theorems Theorem In a tree, there are at least two nodes with degree 1. Proof. 2 E = v V d v assume: only 1 node with degree 1: 2 E 2( V 1) E 2 V 1 E V 1 2 > V 1
20 Tree Theorems Theorem In a tree, there are at least two nodes with degree 1. Proof. 2 E = v V d v assume: only 1 node with degree 1: 2 E 2( V 1) E 2 V 1 E V 1 2 > V 1
21 Tree Theorems Theorem In a tree, there are at least two nodes with degree 1. Proof. 2 E = v V d v assume: only 1 node with degree 1: 2 E 2( V 1) E 2 V 1 E V 1 2 > V 1
22 Tree Theorems Theorem In a tree, there are at least two nodes with degree 1. Proof. 2 E = v V d v assume: only 1 node with degree 1: 2 E 2( V 1) E 2 V 1 E V 1 2 > V 1
23 Tree Theorems Theorem In a tree, there are at least two nodes with degree 1. Proof. 2 E = v V d v assume: only 1 node with degree 1: 2 E 2( V 1) E 2 V 1 E V 1 2 > V 1
24 Topics 1 Trees Introduction Rooted Trees Binary Trees Decision Trees 2 Tree Problems Minimum Spanning Tree
25 Rooted Tree hierarchy between nodes creates implicit direction on edges: in and out degrees in-degree 0: root (only 1 such node) out-degree 0: leaf not a leaf: internal node
26 Rooted Tree hierarchy between nodes creates implicit direction on edges: in and out degrees in-degree 0: root (only 1 such node) out-degree 0: leaf not a leaf: internal node
27 Node Level level of node: distance from root parent: adjacent node closer to root (only 1 such node) child: adjacent nodes further from root sibling: nodes with same parent depth of tree: maximum level in tree
28 Rooted Tree Example root: r leaves: x y z u v internal nodes: r p n t s q w parent of y: w children of w: y and z y and z are siblings
29 Rooted Tree Example Book C1 C2 C3 S1.1 S1.2 S3.1 S3.2 S3.2.1 S3.2.2 S3.3
30 Ordered Rooted Tree siblings ordered from left to right universal address system root: 0 children of root: 1, 2, 3,... v: internal node with address a children of v: a.1, a.2, a.3,...
31 Lexicographic Order address A comes before address B if one of: A = x 1 x 2... x i x j... B = x 1 x 2... x i x k... x j comes before x k A = x 1 x 2... x i B = x 1 x 2... x i x k...
32 Lexicographic Order address A comes before address B if one of: A = x 1 x 2... x i x j... B = x 1 x 2... x i x k... x j comes before x k A = x 1 x 2... x i B = x 1 x 2... x i x k...
33 Lexicographic Order Example
34 Topics 1 Trees Introduction Rooted Trees Binary Trees Decision Trees 2 Tree Problems Minimum Spanning Tree
35 Binary Trees T = (V, E) is a binary tree: v V [d v o {0, 1, 2}] T = (V, E) is a complete binary tree: v V [d v o {0, 2}]
36 Expression Tree binary operations can be represented as binary trees root: operator, children: operands mathematical expression can be represented as trees internal nodes: operators, leaves: variables and values
37 Expression Tree Examples 7 a a + b
38 Expression Tree Examples 7 a 5 (a + b) 3
39 Expression Tree Examples 7 a 5 (a + b) 3
40 Expression Tree Examples t + u v w + x y z
41 Expression Tree Traversals 1 inorder traversal: traverse left subtree, visit root, traverse right subtree 2 preorder traversal: visit root, traverse left subtree, traverse right subtree 3 postorder traversal (reverse Polish notation): traverse left subtree, traverse right subtree, visit root
42 Expression Tree Traversals 1 inorder traversal: traverse left subtree, visit root, traverse right subtree 2 preorder traversal: visit root, traverse left subtree, traverse right subtree 3 postorder traversal (reverse Polish notation): traverse left subtree, traverse right subtree, visit root
43 Expression Tree Traversals 1 inorder traversal: traverse left subtree, visit root, traverse right subtree 2 preorder traversal: visit root, traverse left subtree, traverse right subtree 3 postorder traversal (reverse Polish notation): traverse left subtree, traverse right subtree, visit root
44 Inorder Traversal Example t + u v / w + x y z
45 Preorder Traversal Example + t / u v + w x y z
46 Postorder Traversal Example t u v w x y z + / +
47 Expression Tree Evaluation inorder traversal requires parantheses for precedence preorder and postorder traversals do not require parantheses
48 Postorder Evaluation Example t u v w x y z + / / /
49 Postorder Evaluation Example t u v w x y z + / / /
50 Postorder Evaluation Example t u v w x y z + / / /
51 Postorder Evaluation Example t u v w x y z + / / /
52 Postorder Evaluation Example t u v w x y z + / / /
53 Postorder Evaluation Example t u v w x y z + / / /
54 Postorder Evaluation Example t u v w x y z + / / /
55 Postorder Evaluation Example t u v w x y z + / / /
56 Postorder Evaluation Example t u v w x y z + / / /
57 Postorder Evaluation Example t u v w x y z + / / /
58 Postorder Evaluation Example t u v w x y z + / / /
59 Postorder Evaluation Example t u v w x y z + / / /
60 Postorder Evaluation Example t u v w x y z + / / /
61 Postorder Evaluation Example t u v w x y z + / / /
62 Regular Trees T = (V, E) is an m-ary tree: v V [d v o m] T = (V, E) is a complete m-ary tree: v V [d v o {0, m}]
63 Regular Tree Theorem Theorem T = (V, E): complete m-ary tree n: number of nodes l: number of leaves i: number of internal nodes n = m i + 1 l = n i = m i + 1 i = (m 1) i + 1 i = l 1 m 1
64 Regular Tree Theorem Theorem T = (V, E): complete m-ary tree n: number of nodes l: number of leaves i: number of internal nodes n = m i + 1 l = n i = m i + 1 i = (m 1) i + 1 i = l 1 m 1
65 Regular Tree Theorem Theorem T = (V, E): complete m-ary tree n: number of nodes l: number of leaves i: number of internal nodes n = m i + 1 l = n i = m i + 1 i = (m 1) i + 1 i = l 1 m 1
66 Regular Tree Theorem Theorem T = (V, E): complete m-ary tree n: number of nodes l: number of leaves i: number of internal nodes n = m i + 1 l = n i = m i + 1 i = (m 1) i + 1 i = l 1 m 1
67 Regular Tree Theorem Theorem T = (V, E): complete m-ary tree n: number of nodes l: number of leaves i: number of internal nodes n = m i + 1 l = n i = m i + 1 i = (m 1) i + 1 i = l 1 m 1
68 Regular Tree Theorem Theorem T = (V, E): complete m-ary tree n: number of nodes l: number of leaves i: number of internal nodes n = m i + 1 l = n i = m i + 1 i = (m 1) i + 1 i = l 1 m 1
69 Regular Tree Examples how many matches are played in a tennis tournament with 27 players? every player is a leaf: l = 27 every match is an internal node: m = 2 number of matches: i = l 1 m 1 = = 26
70 Regular Tree Examples how many matches are played in a tennis tournament with 27 players? every player is a leaf: l = 27 every match is an internal node: m = 2 number of matches: i = l 1 m 1 = = 26
71 Regular Tree Examples how many extension cords with 4 outlets are required to connect 25 computers to a wall socket? every computer is a leaf: l = 25 every extension cord is an internal node: m = 4 number of cords: i = l 1 m 1 = = 8
72 Regular Tree Examples how many extension cords with 4 outlets are required to connect 25 computers to a wall socket? every computer is a leaf: l = 25 every extension cord is an internal node: m = 4 number of cords: i = l 1 m 1 = = 8
73 Topics 1 Trees Introduction Rooted Trees Binary Trees Decision Trees 2 Tree Problems Minimum Spanning Tree
74 Decision Trees one of 8 coins is counterfeit (heavier) find counterfeit coin using a beam balance depth of tree: number of weighings
75 Decision Trees
76 Decision Trees
77 Topics 1 Trees Introduction Rooted Trees Binary Trees Decision Trees 2 Tree Problems Minimum Spanning Tree
78 Spanning Tree T = (V, E ) is a spanning tree of G(V, E): T is a subgraph of G T is a tree V = V minimum spanning tree: total weight of edges in E is minimal
79 Kruskal s Algorithm 1 G = (V, E ), V =, E = 2 select e = (v 1, v 2 ) E E such that: E {e} contains no cycle, and wt(e) is minimal 3 E = E {e}, V = V {v 1, v 2 } 4 if E = V 1: result is G 5 go to step 2
80 Kruskal s Algorithm Example minimum weight: 1 (e, g) E = {(e, g)} E = 1
81 Kruskal s Algorithm Example minimum weight: 1 (e, g) E = {(e, g)} E = 1
82 Kruskal s Algorithm Example minimum weight: 1 (e, g) E = {(e, g)} E = 1
83 Kruskal s Algorithm Example minimum weight: 2 (d, e), (d, f ), (f, g) E = {(e, g), (d, f )} E = 2
84 Kruskal s Algorithm Example minimum weight: 2 (d, e), (d, f ), (f, g) E = {(e, g), (d, f )} E = 2
85 Kruskal s Algorithm Example minimum weight: 2 (d, e), (d, f ), (f, g) E = {(e, g), (d, f )} E = 2
86 Kruskal s Algorithm Example minimum weight: 2 (d, e), (f, g) E = {(e, g), (d, f ), (d, e)} E = 3
87 Kruskal s Algorithm Example minimum weight: 2 (d, e), (f, g) E = {(e, g), (d, f ), (d, e)} E = 3
88 Kruskal s Algorithm Example minimum weight: 2 (d, e), (f, g) E = {(e, g), (d, f ), (d, e)} E = 3
89 Kruskal s Algorithm Example minimum weight: 2 (f, g) forms a cycle minimum weight: 3 (c, e), (c, g), (d, g) (d, g) forms a cycle E = {(e, g), (d, f ), (d, e), (c, e)} E = 4
90 Kruskal s Algorithm Example minimum weight: 2 (f, g) forms a cycle minimum weight: 3 (c, e), (c, g), (d, g) (d, g) forms a cycle E = {(e, g), (d, f ), (d, e), (c, e)} E = 4
91 Kruskal s Algorithm Example minimum weight: 2 (f, g) forms a cycle minimum weight: 3 (c, e), (c, g), (d, g) (d, g) forms a cycle E = {(e, g), (d, f ), (d, e), (c, e)} E = 4
92 Kruskal s Algorithm Example minimum weight: 2 (f, g) forms a cycle minimum weight: 3 (c, e), (c, g), (d, g) (d, g) forms a cycle E = {(e, g), (d, f ), (d, e), (c, e)} E = 4
93 Kruskal s Algorithm Example E = { (e, g), (d, f ), (d, e), (c, e), (b, e) } E = 5
94 Kruskal s Algorithm Example E = { (e, g), (d, f ), (d, e), (c, e), (b, e) } E = 5
95 Kruskal s Algorithm Example E = { (e, g), (d, f ), (d, e), (c, e), (b, e), (a, b) } E = 6
96 Kruskal s Algorithm Example E = { (e, g), (d, f ), (d, e), (c, e), (b, e), (a, b) } E = 6
97 Kruskal s Algorithm Example total weight: 17
98 Prim s Algorithm 1 T = (V, E ), E =, v 0 V, V = {v 0 } 2 select v V V such that for a node x V e = (x, v), E {e} contains no cycle, and wt(e) is minimal 3 E = E {e}, V = V {x} 4 if V = V : result is T 5 go to step 2
99 Prim s Algorithm Example E = V = {a} V = 1
100 Prim s Algorithm Example E = V = {a} V = 1
101 Prim s Algorithm Example E = {(a, b)} V = {a, b} V = 2
102 Prim s Algorithm Example E = {(a, b)} V = {a, b} V = 2
103 Prim s Algorithm Example E = {(a, b), (b, e)} V = {a, b, e} V = 3
104 Prim s Algorithm Example E = {(a, b), (b, e)} V = {a, b, e} V = 3
105 Prim s Algorithm Example E = {(a, b), (b, e), (e, g)} V = {a, b, e, g} V = 4
106 Prim s Algorithm Example E = {(a, b), (b, e), (e, g)} V = {a, b, e, g} V = 4
107 Prim s Algorithm Example E = {(a, b), (b, e), (e, g), (d, e)} V = {a, b, e, g, d} V = 5
108 Prim s Algorithm Example E = {(a, b), (b, e), (e, g), (d, e)} V = {a, b, e, g, d} V = 5
109 Prim s Algorithm Example E = { (a, b), (b, e), (e, g), (d, e), (f, g) } V = {a, b, e, g, d, f } V = 6
110 Prim s Algorithm Example E = { (a, b), (b, e), (e, g), (d, e), (f, g) } V = {a, b, e, g, d, f } V = 6
111 Prim s Algorithm Example E = { (a, b), (b, e), (e, g), (d, e), (f, g), (c, g) } V = {a, b, e, g, d, f, c} V = 7
112 Prim s Algorithm Example E = { (a, b), (b, e), (e, g), (d, e), (f, g), (c, g) } V = {a, b, e, g, d, f, c} V = 7
113 Prim s Algorithm Example total weight: 17
114 References Required Reading: Grimaldi Chapter 12: Trees Definitions and Examples Rooted Trees Chapter 13: Optimization and Matching Minimal Spanning Trees: The Algorithms of Kruskal and Prim
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