A word about organization: Since different courses have different lengths of lecture periods, and different instructors go at different paces, rather than dividing the material up into fixed-length lectures, we will divide it up into “modules” which correspond to major topic areas and will generally take 1-3 lectures to cover. Within modules, we have smaller “topics”. Within topics are individual slides.
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Module #22 - Graphs University of Florida Dept. of Computer & Information Science & Engineering COT 3100 Applications of Discrete Structures Dr. Michael P. Frank Slides for a Course Based on the Text Discrete Mathematics & Its Applications (5th Edition) by Kenneth H. Rosen11/06/12 (c)2001-2003, Michael P. Frank 1
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Module #22 - Graphs Module #22: Graph Theory Rosen 5th ed., chs. 8-9 ~44 slides11/06/12 (c)2001-2003, Michael P. Frank 2
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Module #22 - Graphs What are Graphs? Not • General meaning in everyday math: A plot or chart of numerical data using a coordinate system. • Technical meaning in discrete mathematics: A particular class of discrete structures (to be defined) that is useful for representing relations and has a convenient webby- looking graphical representation.11/06/12 (c)2001-2003, Michael P. Frank 3
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Module #22 - Graphs Applications of Graphs • Potentially anything (graphs can represent relations, relations can describe the extension of any predicate). • Apps in networking, scheduling, flow optimization, circuit design, path planning. • More apps: Geneology analysis, computer game-playing, program compilation, object- oriented design, …11/06/12 (c)2001-2003, Michael P. Frank 4
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Module #22 - Graphs Simple Graphs • Correspond to symmetric, irreflexive binary relations R. • A simple graph G=(V,E) Visual Representation consists of: of a Simple Graph – a set V of vertices or nodes (V corresponds to the universe of the relation R), – a set E of edges / arcs / links: unordered pairs of [distinct] elements u,v ∈ V, such that uRv.11/06/12 (c)2001-2003, Michael P. Frank 5
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Module #22 - Graphs Example of a Simple Graph • Let V be the set of states in the far- southeastern U.S.: –I.e., V={FL, GA, AL, MS, LA, SC, TN, NC} • Let E={{u,v}|u adjoins v} TN NC ={{FL,GA},{FL,AL},{FL,MS}, {FL,LA},{GA,AL},{AL,MS}, MS AL SC {MS,LA},{GA,SC},{GA,TN}, GA {SC,NC},{NC,TN},{MS,TN}, LA FL {MS,AL}}11/06/12 (c)2001-2003, Michael P. Frank 6
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Module #22 - Graphs Multigraphs • Like simple graphs, but there may be more than one edge connecting two given nodes. • A multigraph G=(V, E, f ) consists of a set V of vertices, a set E of edges (as primitive objects), and a function Parallel f:E→{{u,v}|u,v∈V ∧ u≠v}. edges • E.g., nodes are cities, edges are segments of major highways.11/06/12 (c)2001-2003, Michael P. Frank 7
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Module #22 - Graphs Pseudographs • Like a multigraph, but edges connecting a node to itself are allowed. (R may be reflexive.) • A pseudograph G=(V, E, f ) where f:E→{{u,v}|u,v∈V}. Edge e∈E is a loop if f(e)={u,u}={u}. • E.g., nodes are campsites in a state park, edges are hiking trails through the woods.11/06/12 (c)2001-2003, Michael P. Frank 8
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Module #22 - Graphs Directed Graphs • Correspond to arbitrary binary relations R, which need not be symmetric. • A directed graph (V,E) consists of a set of vertices V and a binary relation E on V. • E.g.: V = set of People, E={(x,y) | x loves y}11/06/12 (c)2001-2003, Michael P. Frank 9
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Module #22 - Graphs Directed Multigraphs • Like directed graphs, but there may be more than one arc from a node to another. • A directed multigraph G=(V, E, f ) consists of a set V of vertices, a set E of edges, and a function f:E→V×V. • E.g., V=web pages, E=hyperlinks. The WWW is a directed multigraph...11/06/12 (c)2001-2003, Michael P. Frank 10
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Module #22 - Graphs Types of Graphs: Summary • Summary of the book’s definitions. • Keep in mind this terminology is not fully standardized across different authors... Edge Multiple Self- Term type edges ok? loops ok? Simple graph Undir. No No Multigraph Undir. Yes No Pseudograph Undir. Yes Yes Directed graph Directed No Yes Directed multigraph Directed Yes Yes11/06/12 (c)2001-2003, Michael P. Frank 11
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Module #22 - Graphs §8.2: Graph Terminology You need to learn the following terms: • Adjacent, connects, endpoints, degree, initial, terminal, in-degree, out-degree, complete, cycles, wheels, n-cubes, bipartite, subgraph, union.11/06/12 (c)2001-2003, Michael P. Frank 12
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Module #22 - Graphs Adjacency Let G be an undirected graph with edge set E. Let e∈E be (or map to) the pair {u,v}. Then we say: • u, v are adjacent / neighbors / connected. • Edge e is incident with vertices u and v. • Edge e connects u and v. • Vertices u and v are endpoints of edge e.11/06/12 (c)2001-2003, Michael P. Frank 13
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Module #22 - Graphs Degree of a Vertex • Let G be an undirected graph, v∈V a vertex. • The degree of v, deg(v), is its number of incident edges. (Except that any self-loops are counted twice.) • A vertex with degree 0 is called isolated. • A vertex of degree 1 is called pendant.11/06/12 (c)2001-2003, Michael P. Frank 14
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Module #22 - Graphs Handshaking Theorem • Let G be an undirected (simple, multi-, or pseudo-) graph with vertex set V and edge set E. Then ∑ deg(v) = 2 E v∈V • Corollary: Any undirected graph has an even number of vertices of odd degree.11/06/12 (c)2001-2003, Michael P. Frank 15
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Module #22 - Graphs Directed Adjacency • Let G be a directed (possibly multi-) graph, and let e be an edge of G that is (or maps to) (u,v). Then we say: – u is adjacent to v, v is adjacent from u – e comes from u, e goes to v. – e connects u to v, e goes from u to v – the initial vertex of e is u – the terminal vertex of e is v11/06/12 (c)2001-2003, Michael P. Frank 16
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Module #22 - Graphs Directed Degree • Let G be a directed graph, v a vertex of G. – The in-degree of v, deg−(v), is the number of edges going to v. – The out-degree of v, deg+(v), is the number of edges coming from v. – The degree of v, deg(v):≡deg−(v)+deg+(v), is the sum of v’s in-degree and out-degree.11/06/12 (c)2001-2003, Michael P. Frank 17
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Module #22 - Graphs Directed Handshaking Theorem • Let G be a directed (possibly multi-) graph with vertex set V and edge set E. Then: 1 ∑ δεγ (v) = ∑ δεγ (v) = 2 ∑ δεγ( v) = E v∈V − v∈V + v∈V • Note that the degree of a node is unchanged by whether we consider its edges to be directed or undirected.11/06/12 (c)2001-2003, Michael P. Frank 18
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Module #22 - Graphs Special Graph Structures Special cases of undirected graph structures: • Complete graphs Kn • Cycles Cn • Wheels Wn • n-Cubes Qn • Bipartite graphs • Complete bipartite graphs Km,n11/06/12 (c)2001-2003, Michael P. Frank 19
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Module #22 - Graphs Complete Graphs • For any n∈N, a complete graph on n vertices, Kn, is a simple graph with n nodes in which every node is adjacent to every other node: ∀u,v∈V: u≠v↔{u,v}∈E. K1 K2 K3 K4 K5 K6 ν( ν − 1) ν −1 Note that Kn has ∑ ι = 2 edges. ι=111/06/12 (c)2001-2003, Michael P. Frank 20
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Module #22 - Graphs Cycles • For any n≥3, a cycle on n vertices, Cn, is a simple graph where V={v1,v2,… ,vn} and E={{v1,v2},{v2,v3},…,{vn−1,vn},{vn,v1}}. C3 C4 C5 C6 C8 C7 How many edges are there in Cn?11/06/12 (c)2001-2003, Michael P. Frank 21
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Module #22 - Graphs Wheels • For any n≥3, a wheel Wn, is a simple graph obtained by taking the cycle Cn and adding one extra vertex vhub and n extra edges {{vhub,v1}, {vhub,v2},…,{vhub,vn}}. W3 W4 W5 W6 W8 W7 How many edges are there in Wn?11/06/12 (c)2001-2003, Michael P. Frank 22
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Module #22 - Graphs n-cubes (hypercubes) • For any n∈N, the hypercube Qn is a simple graph consisting of two copies of Qn-1 connected together at corresponding nodes. Q0 has 1 node. Q0 Q1 Q2 Q4 Q3 Number of vertices: 2n. Number of edges:Exercise to try!11/06/12 (c)2001-2003, Michael P. Frank 23
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Module #22 - Graphs n-cubes (hypercubes) • For any n∈N, the hypercube Qn can be defined recursively as follows: – Q0={{v0},∅} (one node and no edges) – For any n∈N, if Qn=(V,E), where V={v1,…,va} and E={e1,…,eb}, then Qn+1=(V∪{v1´,…,va´}, E∪{e1´,…,eb´}∪{{v1,v1´},{v2,v2´},…, {va,va´}}) where v1´,…,va´ are new vertices, and where if ei={vj,vk} then ei´={vj´,vk´}.11/06/12 (c)2001-2003, Michael P. Frank 24
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Module #22 - Graphs Bipartite Graphs • Def’n.: A graph G=(V,E) is bipartite (two- part) iff V = V1∩V2 where V1∪V2=∅ and ∀e∈E: ∃v1∈V1,v2∈V2: e={v1,v2}. • In English: The graph can be divided into two parts in such a way that all edges go between the two parts. V1 V2 This definition can easily be adapted for the case of multigraphs and directed graphs as well. Can represent with zero-one matrices.11/06/12 (c)2001-2003, Michael P. Frank 25
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Module #22 - Graphs Complete Bipartite Graphs • For m,n∈N, the complete bipartite graph Km,n is a bipartite graph where |V1| = m, |V2| = n, and E = {{v1,v2}|v1∈V1 ∧ v2∈V2}. – That is, there are m nodes K4,3 in the left part, n nodes in the right part, and every node in the left part is connected to every node Km,n has _____ nodes in the right part. and _____ edges.11/06/12 (c)2001-2003, Michael P. Frank 26
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Module #22 - Graphs Subgraphs • A subgraph of a graph G=(V,E) is a graph H=(W,F) where W⊆V and F⊆E. G H11/06/12 (c)2001-2003, Michael P. Frank 27
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Module #22 - Graphs Graph Unions • The union G1∪G2 of two simple graphs G1=(V1, E1) and G2=(V2,E2) is the simple graph (V1∪V2, E1∪E2). ∪ a b c a b c d e d f11/06/12 (c)2001-2003, Michael P. Frank 28
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Module #22 - Graphs §8.3: Graph Representations & Isomorphism • Graph representations: – Adjacency lists. – Adjacency matrices. – Incidence matrices. • Graph isomorphism: – Two graphs are isomorphic iff they are identical except for their node names.11/06/12 (c)2001-2003, Michael P. Frank 29
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Module #22 - Graphs Adjacency Lists • A table with 1 row per vertex, listing its adjacent vertices. Adjacent a b Vertex Vertices a b, c c d b a, c, e, f e f c a, b, f d e b f c, b11/06/12 (c)2001-2003, Michael P. Frank 30
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Module #22 - Graphs Directed Adjacency Lists • 1 row per node, listing the terminal nodes of each edge incident from that node.11/06/12 (c)2001-2003, Michael P. Frank 31
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Module #22 - Graphs Adjacency Matrices • A way to represent simple graphs – possibly with self-loops. • Matrix A=[aij], where aij is 1 if {vi, vj} is an edge of G, and is 0 otherwise. • Can extend to pseudographs by letting each matrix elements be the number of links (possibly >1) between the nodes.11/06/12 (c)2001-2003, Michael P. Frank 32
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Module #22 - Graphs Graph Isomorphism • Formal definition: – Simple graphs G1=(V1, E1) and G2=(V2, E2) are isomorphic iff ∃ a bijection f:V1→V2 such that ∀ a,b∈V1, a and b are adjacent in G1 iff f(a) and f(b) are adjacent in G2. – f is the “renaming” function between the two node sets that makes the two graphs identical. – This definition can easily be extended to other types of graphs.11/06/12 (c)2001-2003, Michael P. Frank 33
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Module #22 - Graphs Graph Invariants under Isomorphism Necessary but not sufficient conditions for G1=(V1, E1) to be isomorphic to G2=(V2, E2): – We must have that |V1|=|V2|, and |E1|=|E2|. – The number of vertices with degree n is the same in both graphs. – For every proper subgraph g of one graph, there is a proper subgraph of the other graph that is isomorphic to g.11/06/12 (c)2001-2003, Michael P. Frank 34
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Module #22 - Graphs Isomorphism Example • If isomorphic, label the 2nd graph to show the isomorphism, else identify difference. b d a b a d c e e c f f11/06/12 (c)2001-2003, Michael P. Frank 35
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Module #22 - Graphs Are These Isomorphic? • If isomorphic, label the 2nd graph to show the isomorphism, else identify difference. • Same # of a vertices b • Same # of edges • Different # of d verts of c e degree 2! (1 vs 3)11/06/12 (c)2001-2003, Michael P. Frank 36
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Module #22 - Graphs §8.4: Connectivity • In an undirected graph, a path of length n from u to v is a sequence of adjacent edges going from vertex u to vertex v. • A path is a circuit if u=v. • A path traverses the vertices along it. • A path is simple if it contains no edge more than once.11/06/12 (c)2001-2003, Michael P. Frank 37
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Module #22 - Graphs Paths in Directed Graphs • Same as in undirected graphs, but the path must go in the direction of the arrows.11/06/12 (c)2001-2003, Michael P. Frank 38
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Module #22 - Graphs Connectedness • An undirected graph is connected iff there is a path between every pair of distinct vertices in the graph. • Theorem: There is a simple path between any pair of vertices in a connected undirected graph. • Connected component: connected subgraph • A cut vertex or cut edge separates 1 connected component into 2 if P. Frank 3911/06/12 (c)2001-2003, Michael removed.
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Module #22 - Graphs Directed Connectedness • A directed graph is strongly connected iff there is a directed path from a to b for any two verts a and b. • It is weakly connected iff the underlying undirected graph (i.e., with edge directions removed) is connected. • Note strongly implies weakly but not vice- versa.11/06/12 (c)2001-2003, Michael P. Frank 40
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Module #22 - Graphs Paths & Isomorphism • Note that connectedness, and the existence of a circuit or simple circuit of length k are graph invariants with respect to isomorphism.11/06/12 (c)2001-2003, Michael P. Frank 41
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Module #22 - Graphs Counting Paths w Adjacency Matrices • Let A be the adjacency matrix of graph G. • The number of paths of length k from vi to vj is equal to (Ak)i,j. – The notation (M)i,j denotes mi,j where [mi,j] = M.11/06/12 (c)2001-2003, Michael P. Frank 42
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Module #22 - Graphs §8.5: Euler & Hamilton Paths • An Euler circuit in a graph G is a circuit containing every edge of G. • An Euler path in G is a path containing every edge of G. • A Hamilton circuit is a circuit that traverses each vertex in G exactly once. • A Hamilton path is a path that traverses each vertex in G exactly once.11/06/12 (c)2001-2003, Michael P. Frank 43
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Module #22 - Graphs Bridges of Königsberg Problem • Can we walk through town, crossing each bridge exactly once, and return to start? A B D C Equivalent multigraph The original problem11/06/12 (c)2001-2003, Michael P. Frank 44
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Module #22 - Graphs Euler Path Theorems • Theorem: A connected multigraph has an Euler circuit iff each vertex has even degree. – Proof: • (→) The circuit contributes 2 to degree of each node. • (←) By construction using algorithm on p. 580-581 • Theorem: A connected multigraph has an Euler path (but not an Euler circuit) iff it has exactly 2 vertices of odd degree. – One is the start, the other is the end.11/06/12 (c)2001-2003, Michael P. Frank 45
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Module #22 - Graphs Euler Circuit Algorithm • Begin with any arbitrary node. • Construct a simple path from it till you get back to start. • Repeat for each remaining subgraph, splicing results back into original cycle.11/06/12 (c)2001-2003, Michael P. Frank 46
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Module #22 - Graphs Round-the-World Puzzle • Can we traverse all the vertices of a dodecahedron, visiting each once?` Equivalent Dodecahedron puzzle Pegboard version graph11/06/12 (c)2001-2003, Michael P. Frank 47
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Module #22 - Graphs Hamiltonian Path Theorems • Dirac’s theorem: If (but not only if) G is connected, simple, has n≥3 vertices, and ∀v deg(v)≥n/2, then G has a Hamilton circuit. – Ore’s corollary: If G is connected, simple, has n≥3 nodes, and deg(u)+deg(v)≥n for every pair u,v of non-adjacent nodes, then G has a Hamilton circuit.11/06/12 (c)2001-2003, Michael P. Frank 48
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Module #22 - Graphs HAM-CIRCUIT is NP-complete • Let HAM-CIRCUIT be the problem: – Given a simple graph G, does G contain a Hamiltonian circuit? • This problem has been proven to be NP-complete! – This means, if an algorithm for solving it in polynomial time were found, it could be used to solve all NP problems in polynomial time.11/06/12 (c)2001-2003, Michael P. Frank 49
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Module #22 - Graphs §9.1: Introduction to Trees • A tree is a connected undirected graph that contains no circuits. – Theorem: There is a unique simple path between any two of its nodes. • A (not-necessarily-connected) undirected graph without simple circuits is called a forest. – You can think of it as a set of trees having disjoint sets of nodes. • A leaf node in a tree or forest is any pendant or isolated vertex. An internal node is any non-leaf vertex (thus it has degree ≥ ___ ).11/06/12 (c)2001-2003, Michael P. Frank 50
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Module #22 - Graphs Tree and Forest Examples Leaves in green, internal nodes in brown. • A Tree: • A Forest:11/06/12 (c)2001-2003, Michael P. Frank 51
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Module #22 - Graphs Rooted Trees • A rooted tree is a tree in which one node has been designated the root. – Every edge is (implicitly or explicitly) directed away from the root. • You should know the following terms about rooted trees: – Parent, child, siblings, ancestors, descendents, leaf, internal node, subtree.11/06/12 (c)2001-2003, Michael P. Frank 52
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Module #22 - Graphs Rooted Tree Examples • Note that a given unrooted tree with n nodes yields n different rooted trees. Same tree except root for choice of root root11/06/12 (c)2001-2003, Michael P. Frank 53
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Module #22 - Graphs Rooted-Tree Terminology Exercise • Find the parent, o h n r children, siblings, ancestors, & d m b descendants root of node f. a c g e i q f p j k l11/06/12 (c)2001-2003, Michael P. Frank 54
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Module #22 - Graphs n-ary trees • A rooted tree is called n-ary if every vertex has no more than n children. – It is called full if every internal (non-leaf) vertex has exactly n children. • A 2-ary tree is called a binary tree. – These are handy for describing sequences of yes-no decisions. • Example: Comparisons in binary search algorithm.11/06/12 (c)2001-2003, Michael P. Frank 55
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Module #22 - Graphs Which Tree is Binary? • Theorem: A given rooted tree is a binary tree iff every node other than the root has degree ≤ ___, and the root has degree ≤ ___.11/06/12 (c)2001-2003, Michael P. Frank 56
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Module #22 - Graphs Ordered Rooted Tree • This is just a rooted tree in which the children of each internal node are ordered. • In ordered binary trees, we can define: – left child, right child – left subtree, right subtree • For n-ary trees with n>2, can use terms like “leftmost”, “rightmost,” etc.11/06/12 (c)2001-2003, Michael P. Frank 57
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Module #22 - Graphs Trees as Models • Can use trees to model the following: – Saturated hydrocarbons – Organizational structures – Computer file systems • In each case, would you use a rooted or a non-rooted tree?11/06/12 (c)2001-2003, Michael P. Frank 58
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Module #22 - Graphs Some Tree Theorems • Any tree with n nodes has e = n−1 edges. – Proof: Consider removing leaves. • A full m-ary tree with i internal nodes has n=mi+1 nodes, and =(m−1)i+1 leaves. – Proof: There are mi children of internal nodes, plus the root. And, = n−i = (m−1)i+1. □ • Thus, when m is known and the tree is full, we can compute all four of the values e, i, n, and , given any one of them.11/06/12 (c)2001-2003, Michael P. Frank 59
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Module #22 - Graphs Some More Tree Theorems • Definition: The level of a node is the length of the simple path from the root to the node. – The height of a tree is maximum node level. – A rooted m-ary tree with height h is called balanced if all leaves are at levels h or h−1. • Theorem: There are at most mh leaves in an m-ary tree of height h. – Corollary: An m-ary tree with leaves has height h≥logm . If m is full and balanced then h=logm.11/06/12 (c)2001-2003, Michael P. Frank 60
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Module #22 - Graphs §9.2: Applications of Trees • Binary search trees – A simple data structure for sorted lists • Decision trees – Minimum comparisons in sorting algorithms • Prefix codes – Huffman coding • Game trees11/06/12 (c)2001-2003, Michael P. Frank 61
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Module #22 - Graphs Binary Search Trees • A representation for sorted sets of items. – Supports the following operations in Θ(log n) average-case time: • Searching for an existing item. • Inserting a new item, if not already present. – Supports printing out all items in Θ(n) time. • Note that inserting into a plain sequence ai would instead take Θ(n) worst-case time.11/06/12 (c)2001-2003, Michael P. Frank 62
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Module #22 - Graphs Binary Search Tree Format • Items are stored at Example: individual tree nodes. 7 • We arrange for the tree to always obey 3 12 this invariant: – For every item x, • Every node in x’s left 1 5 9 15 subtree is less than x. • Every node in x’s right subtree is greater than x. 0 2 8 1111/06/12 (c)2001-2003, Michael P. Frank 63
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Module #22 - Graphs Recursive Binary Tree Insert procedure insert(T: binary tree, x: item) v := root[T] if v = null then begin root[T] := x; return “Done” end else if v = x return “Already present” else if x < v then return insert(leftSubtree[T], x) else {must be x > v} return insert(rightSubtree[T], x)11/06/12 (c)2001-2003, Michael P. Frank 64
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Module #22 - Graphs Decision Trees (pp. 646-649) • A decision tree represents a decision-making process. – Each possible “decision point” or situation is represented by a node. – Each possible choice that could be made at that decision point is represented by an edge to a child node. • In the extended decision trees used in decision analysis, we also include nodes that represent random events and their outcomes.11/06/12 (c)2001-2003, Michael P. Frank 65
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Module #22 - Graphs Coin-Weighing Problem • Imagine you have 8 coins, one of which is a lighter counterfeit, and a free-beam balance. – No scale of weight markings is required for this problem! • How many weighings are ? needed to guarantee that the counterfeit coin will be found?11/06/12 (c)2001-2003, Michael P. Frank 66
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Module #22 - Graphs As a Decision-Tree Problem • In each situation, we pick two disjoint and equal- size subsets of coins to put on the scale.The balance then A given sequence of“decides” whether weighings thus yieldsto tip left, tip right, a decision tree withor stay balanced. branching factor 3.11/06/12 (c)2001-2003, Michael P. Frank 67
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Module #22 - Graphs Applying the Tree Height Theorem • The decision tree must have at least 8 leaf nodes, since there are 8 possible outcomes. – In terms of which coin is the counterfeit one. • Recall the tree-height theorem, h≥logm. – Thus the decision tree must have height h ≥ log38 = 1.893… = 2. • Let’s see if we solve the problem with only 2 weighings…11/06/12 (c)2001-2003, Michael P. Frank 68
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Module #22 - Graphs General Solution Strategy • The problem is an example of searching for 1 unique particular item, from among a list of n otherwise identical items. – Somewhat analogous to the adage of “searching for a needle in haystack.” • Armed with our balance, we can attack the problem using a divide- and-conquer strategy, like what’s done in binary search. – We want to narrow down the set of possible locations where the desired item (coin) could be found down from n to just 1, in a logarithmic fashion. • Each weighing has 3 possible outcomes. – Thus, we should use it to partition the search space into 3 pieces that are as close to equal-sized as possible. • This strategy will lead to the minimum possible worst-case number of weighings required.11/06/12 (c)2001-2003, Michael P. Frank 69
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Module #22 - Graphs General Balance Strategy • On each step, put n/3 of the n coins to be searched on each side of the scale. – If the scale tips to the left, then: • The lightweight fake is in the right set of n/3 ≈ n/3 coins. – If the scale tips to the right, then: • The lightweight fake is in the left set of n/3 ≈ n/3 coins. – If the scale stays balanced, then: • The fake is in the remaining set of n − 2n/3 ≈ n/3 coins that were not weighed! • Except if n mod 3 = 1 then we can do a little better by weighing n/3 of the coins on each side. You can prove that this strategy always leads to a balanced 3-ary tree.11/06/12 (c)2001-2003, Michael P. Frank 70
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Module #22 - Graphs Coin Balancing Decision Tree • Here’s what the tree looks like in our case: left: 123 vs 456 123 balanced: right: 78 456 1 vs. 2 4 vs. 5 7 vs. 8 L:1 R:2 B:3 L:4 R:5 B:6 L:7 R:811/06/12 (c)2001-2003, Michael P. Frank 71
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Module #22 - Graphs §9.3: Tree Traversal • Universal address systems • Traversal algorithms – Depth-first traversal: • Preorder traversal • Inorder traversal • Postorder traversal – Breadth-first traversal • Infix/prefix/postfix notation11/06/12 (c)2001-2003, Michael P. Frank 72
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