Discrete Mathematics. Chapter 7. trees Sanguk Noh

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1 Discrete Mathematics Chapter 7. trees Sanguk Noh

2 Table Trees Labeled Trees Tree searching Undirected trees Minimal Spanning Trees

3 Trees Theorem : Let (T, v ) be a rooted tree. Then, There are no cycles in T. v is the only root of T. Each vertex in T, other than v, has in-degree one. And v has indegree zero.

4 Trees Notation No edges enter v. v The root Level parent v v 2 v 3 Level height of the tree v 4 v 5 v 6 v 7 v 8 Level 2 offspring leaves (have no offspring) Siblings(the offspring of any one vertex) * ordered tree : some ordering at each level by arranging offspring from left to right

5 Trees Theorem : : Let (T, v ) be a rooted tree on a set A T is irreflexive. T is asymmetric. If (a,b) T and (b,c) T, then (a,c) T for all a,b,c in A Notation v At most n offspring : n-tree v v 2 v 3 All vertices of T have exactly n offspring Complete n-tree * 2-tree : binary tree

6 T(v 6 ) Trees Theorem If (T, v ) is a rooted tree and v T, then T(v) is also a rooted tree with root v. T(v): the subtree of T beginning at v. e.g.) A={v,v 2,.,v } T={(v 2, v 3 ), (v 2, v ), (v 4, v 5 ), (v 4, v 6 ), (v 5, v 8 ), (v 6, v 7 ), (v 4, v 2 ), (v 7, v 9 ), (v 7, v )} v 4 (T,v 4 ): the tree v 6 v 2 v 5 T(v 5 ) v 7 v v 3 v 8 v 9 v T(v 2 )

7 Labeled Trees The fully parenthesized, algebraic expression (3-(2*x)) + ((x-2)-(3+x)) + Central operator * x x 2 3 x *tree data structure. n-tree(t,v )

8 Labeled Trees Computer representation of binary trees Doubly linked list. (searching a set of data in either direction) Left pointer data Data storage Right pointer Pointer to the next cell. (an address where the next cell is located) For a pointer specifying no additional date. (: the corresponding off spring does not exist)

9 Labeled Trees e.g.) start x Fig.7. p277 Index Left Data Right * 2 x

10 Tree searching Tree search : the process of visiting each vertex of a tree in some specific order Positional binary tree(labeled digraph) (we place in its appropriate position each offspring that actually occurs.) The positions for potential offspring are labeled left and right. L L R R * T : a binary positional tree with root v. (T,v) T(V L ) : the left subtree of T T(V R ): the right subtree of T R L R

11 Tree searching Preorder search B A H Algorithm PREORDER(T,v) print v // visit the root if v L NIL then PREORDER(T(v L ),V L ) //search the left subtree if v R NIL then PREORDER(T(v R ),V R ) //search the right subtree end PREORDER C E I K D F G J L The result of the complete search of T A B A B C A B C E D C Nil D Nil A B C D D F G ABCDEFG I H K A B C D E A B C D E F A B C D E F G F G Nil Nil Nil Nil J L

12 Tree searching e.g.) the prefix of the given algebraic expression by applying PREORDER to the tree * (a-b)*((c+(d/e)) * - * - a b a b c / a b c / d e d e Unambiguous expression w/o parentheses *-ab+c/de Move from left to right F xy F: symbol (+,-,*,/,etc) x&y: numbers Continue this procedure until only one number remains. Ex) *-64+5/22 =>

13 Tree searching Inorder search Algorithm INORDER(T,v) if v L NIL then INORDER(T(v L ),V L ) //search the left subtree print v // visit the root if v R NIL then INORDER(T(v R ),V R )//search the right subtree end INORDER Postorder search Algorithm POSTORDER(T,v) if v L NIL then POSTORDER(T(v L ),V L ) //search the left subtree if v R NIL then POSTORDER(T(v R ),V R ) //search the right subtree print v // visit the root end POSTORDER

14 Tree searching A B H *Inorder search A B C D D C B E C E I K D nil nil F G D F G J L DCBFEGA. *Postorder search A B C D C B E D F E G F G DCFGEB.

15 Tree searching e.g.) algebraic expression a - b + c /. infix a-b*c+d/e : ambiguous! 2. postfix ab-cde/+* a=6, b=4, c=5, d=2, e= /+* 2 5 +* 2 6* 2 unambiguous the method of evaluating expressions in some calculations d e

16 Tree searching Searching General Trees T : ordered tree A : the set of vertices of T B(T) : binary positional tree If v A, then the left offspring v L of v in B(T) the first offspring of v in T the right offspring v R of v in B(T) the next sibling of v in T e.g.) general tree: T corresponding binding positional tree: B(T)

17 Tree searching 7.3 Problem #2 Fig preorder - inorder - postorder

18 Undirected trees Def.) Undirected tree: the symmetric closure of a tree (bidirectional edges) Undirected edge of T The set {a,b}, where (a,b) and (b,a) are in T. (a, b: adjacent vertices) Def.) spanning tree R : a connected, undirected relation on A T : a spanning tree of R If A vertices and A - edges, then the edges connect all the vertices.

19 Undirected trees e.g.) a connected, undirected relation R a b c d e f Spanning trees (not unique!) a b c d e f b a c d e f undirected spanning tree a b c d e f

20 Undirected trees Algorithm Merging-process // R : a relation on a set A // a, b A while (R is not undirected spanning tree) A =A-{a,b} A =A {a }, a A // R on A // u,v A, where u a, v a (a, u) R iff (a,u) R or (b,u) R (u, a ) R iff (u,a) R or (u,b) R (u, v) R iff (u,v) R end while End Merging-process

21 Undirected trees e.g.) a symmetric relation v v v 2 v v 2 v v 5 v 5 v 3 v 4 v 5 v 6 v 3 v 4 v 6 v 3 v 4 v 6 * Merging process {v, v } A ={v 2, v 3, v 4, v 5, v 6 } A = A {v } ={v, v 2, v 3, v 4, v 5, v 6 } {v, v 2 } A ={v 3, v 4, v 5, v 6 } A = A {v } ={v, v 3, v 4, v 5, v 6 }

22 Undirected trees Matrix of R (merging two vertices into a new vertex) Step. row i-vertex a row j-vertex b replace row i by the join (either has a ) of rows i and j Step 2. Replace column i by the join of columns i and j Step 3. Restore the main diagonal to its original values in R (avoid cycle of length ) Step 4. Delete row j and column j

23 Undirected trees v v 2 v v v 5 v 2 v v 2 v 3 v 4 v 5 v 6 v v 2 v 3 v 4 v 5 v 6 v 3 v 4 v 5 v 6 v 3 v 4 v 6 v v v 2 v 3 v 4 v 5 v 6 v v v 2 v 3 v 4 v 5 v 6 v v 3 v 4 v 5 v 6 v v 3 v 4 v 5 v 6 v v 3 v 4 v 5 v 6

24 Undirected trees Algorithm Prim //finding a spanning tree/ Do Choose v ( R) and v is the st row in the matrix. Choose v 2 s.t. (v, v 2 ) R. Merge v &v 2 into a new vertex v. Compute the matrix of the resulting relation R. v {v, v 2 } ST ST {(v, v 2 )} Until ( A =) // Single vertex is obtained End Prim

25 Undirected trees e.g.) a c b <by deleting some edges of R> a b root d c d a b c d a b c d Merged vertices - New vertex c (a,c) a=root a b d a b d a {a,c} b (c,b) or (a,b) a d a a a a a {a,c,b} a {a,c,b,d} d - (a,d) or (c,d) or (b,d)

26 Minimal Spanning Trees Minimal Spanning Trees Def.) weighted graph each edge is labeled with a numerical value (weight). Def.) nearest neighbor of vertex if vertices u and v are adjacent and the weight on the edge is minimum. Def.) minimal spanning tree undirected spanning tree for which the total weight of the edges in the tree is as small as possible.

27 Minimal Spanning Trees Algorithm MST-Prim // R with n vertices V {v } E { } Do v j V v i nearest-neighbor(v j ) (v i, v j ) does not form a cycle with members of E. // whether the edge is safe or not V V {v i } E E {(v i, v j )} Until ( E =n-) End MST-Prim

28 Minimal Spanning Trees The greedy strategy advocates making the choice that is the best at the moment. e.g.) A 3 C B E D H 6 A B D H E 2 C 3 F 4 G F 4 G The root vertex : A

29 Minimal Spanning Trees Algorithm MST-Kruskal // R with n vertices // S={e,e 2, e k }: the set of weighted edges // e of least weight E {e } S S-{e } Do Select e i S of least weight that will not make a cycle. E E {e i } S S-{e i } Until ( E =n-) End MST-Kruskal

30 Minimal Spanning Trees e.g.) Figure 7.49 & Fig Mst-Kruskal Mst-Prim

Chapter 4 Trees. Theorem A graph G has a spanning tree if and only if G is connected.

Chapter 4 Trees. Theorem A graph G has a spanning tree if and only if G is connected. Chapter 4 Trees 4-1 Trees and Spanning Trees Trees, T: A simple, cycle-free, loop-free graph satisfies: If v and w are vertices in T, there is a unique simple path from v to w. Eg. Trees. Spanning trees: