DATA STRUCTURES
`
`ehaL
`
`JOHN E. HOPCRO
`Saltmeme)-\,|
`
`
`
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`

`

`Data Structures
`
`and Algorithms
`
`ALFRED V. AHO
`
`Bell Laboratories
`Murray Hill, New Jersey
`
`JOHN E. HOPCROFT
`
`Cornell University
`Ithaca, New York
`
`JEFFREY D. ULLMAN
`
`Stanford University
`Stanford, California
`
`.,•., ADDISON-WESLEY PUBLISHING COMPANY
`Reading, Massachusetts • Menlo Park, California
`London • Amsterdam • Don Mills, Ontario • Sydney
`
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`

`

`This book is in the
`ADDISON-WESLEY SERIES IN
`COMPUTER SCIENCE AND INFORMATION PROCESSING
`
`MichaeJ A. Harrison
`Consulting Editor
`
`Ubnry of Congress C.liloging In PubUcatlon Data
`
`Aho. Alfred V.
`Data structures and algorithms.
`1. Data structures (Computer science) 2. Algorithms.
`I. Hopcroft, John E., 1939-
`. II. Ullman,
`Jeffrey D., 1942-
`III. Title.
`QA76.9.D35A38 1982
`001.64
`ISBN (}.201-00023-7
`
`82-11596
`
`Reproduced by Addison-Wesley from camera-ready copy supplied by the authors.
`
`Reprinted with corrections April, 1987
`Copyright e 1983 by Bell Telephone Laboratories, Incorporated.
`
`All rights reserved. No part of this publication may be reproduced, stored in a re(cid:173)
`trieval system, or transmitted, in any form or by any means, electronic, mechanical,
`photocopying, recording, or otherwise, without the prior written permission of the pub(cid:173)
`lisher. Printed in the United States of America. Published simultaneously in Canada.
`
`ISBN: 0-201-00023-7
`
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`Contents
`
`Chapter 1
`
`I. I
`1.2
`l.3
`l.4
`l.5
`1.6
`l.7
`
`Chapter 2
`
`2. l
`2.2
`2.3
`2.4
`2.5
`2.6
`
`Chapter 3
`
`3. l
`3.2
`3.3
`3.4
`
`Chapter 4
`
`4. l
`4.2
`4.3
`4.4
`4.5
`4.6
`4.7
`4.S
`4.9
`4.10
`4.ll
`4.12
`
`Design and Analysis of Algorithms
`From Problems to Programs ................................................ I
`Abstract Data Types ........................................................ 10
`Data Types, Data Structures, and Abstract Data Types ............ 13
`The Running Time of a Program ........................................ 16
`Calculating the Running Time of a Program .......................... 21
`Good Programming Practice .............................................. 27
`Super Pascal ................................................................... 29
`
`Basic Data Types
`The Data Type "List" ...................................................... 37
`Implementation of Lists ................................................... .40
`Stacks ........................................................................... 53
`Queues ......................................................................... 56
`Mappings ...................................................................... 61
`Stacks and Recursive Procedures ........................................ 64
`
`Trees
`Basic Terminology ........................................................... 75
`The ADT TREE ............................................................. 82
`Implementations of Trees .................................................. 84
`Binary Trees .................................................................. 93
`
`Basic Operations on Sets
`Introduction to Sets ........................................................ I 07
`An ADT with Union, Intersection, and Difference ................ 109
`A Bit-Vector Implementation of Sets .................................. 112
`A Linked-List Implementation of Sets ................................ 115
`The Dictionary .............................................................. 117
`Simple Dictionary Implementations .................................... 119
`The Hash Table Data Structure ......................................... 122
`Estimating the Efficiency of Hash Functions ........................ 129
`Implementation of the Mapping ADT ................................. 135
`Priority Queues ............................................................. 135
`Implementations of Priority Queues ................................... 138
`Some Complex Set Structures ........................................... 145
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`x
`
`CONTENTS
`
`Chapter 5
`
`Advanced Set Representation Methods
`
`5.1
`5.2
`5.3
`5.4
`5.5
`5.6
`
`Binary Search Trees ....................................................... 155
`Time Analysis of Binary Search Tree Operations .................. 160
`Tries ........................................................................... 163
`Balanced Tree Implementations of Sets ............................... 169
`Sets with the MERGE and FIND Operations ....................... 180
`An ADT with MERGE and SPLIT .................................... 189
`
`Chapter 6
`
`Directed Graphs
`
`6.1
`6.2
`6.3
`6.4
`6.5
`6.6
`6.7
`
`Basic Definitions .................................................... , ...... 198
`Representations for Directed Graphs ..... , ... , ........................ 199
`The Single-Source Shortest Paths Problem ........................... 203
`The All-Pairs Shortest Path Problem .................................. 208
`Traversals of Directed Graphs .......................................... 215
`Directed Acyclic Graphs .................................................. 219
`Strong Components ........................................................ 222
`
`Chapter 7
`
`Undirected Graphs
`
`7.1
`7.2
`7.3
`7.4
`7.5
`Chapter 8
`
`8.1
`8.2
`8.3
`8.4
`8.5
`8.6
`8.7
`Chapter 9
`
`Definitions . . . . . . . . . . . . . ... , ................................................. 230
`Minimum-Cost Spanning Trees ......................................... 233
`Traversals .......................................................... , ......... 239
`Articulation Points and Biconnected Components .................. 244
`Graph Matching ......................................... , ................. , 246
`
`Sorting
`
`The Internal Sorting Model. ............. , ............................... 253
`Some Simple Sorting Schemes ................................ , .......... 254
`Quicksort ..................................................................... 260
`Heapsort ...................................................................... 271
`Bin Sorting ................................................................... 274
`A Lower Bound for Sorting by Comparisons ........................ 282
`Order Statistics ........ ,, .................................................... 286
`
`Algorithm Analysis Techniques
`
`9.1
`9.2
`9.3
`9.4
`
`Efficiency of Algorithms ................................................. 293
`Analysis of Recursive Programs ..... , .................................. 294
`Solving Recurrence Equations .......................................... 296
`A General Solution for a Large Class of Recurrences ...... , ..... 298
`
`Chapter 10
`IO.I
`10.2
`10.3
`10.4
`10.5
`
`Algorithm Design Techniques
`
`Divide-and-Conquer Algorithms ........................................ 306
`Dynamic Programming ................................................... 311
`Greedy Algorithms ........................................................ 321
`Backtracking ................................................................. 324
`Local Search Algorithms ................................................. 336
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`xi
`
`CONTENTS
`
`Chapter 11
`
`11.1
`11.2
`11.3
`11.4
`
`Chapter 12
`12. l
`12.2
`12.3
`12.4
`12.5
`12.6
`
`Data Structures and Algorithms for External Storage
`
`A Model of External Computation ..................................... 347
`External Sorting ............................................................ 349
`Storing Information in Files ............................................. 361
`External Search Trees ..................................................... 368
`
`Memory Management
`The Issues in Memory Management ................................... 378
`Managing Equal-Sized Blocks ........................................... 382
`Garbage Collection Algorithms for Equal-Sized Blocks .......... 384
`Storage Allocation for Objects with Mixed Sizes ................... 392
`Buddy Systems ............................................................. .400
`Storage Compaction ...................................................... .404
`
`Bibliography ............................................................... .411
`
`Index ......................................................................... .419
`
`~~~----------------------·
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`CHAPTER 3
`
`Trees
`
`A tree imposes a hierarchical structure on a collection of items. Familiar
`examples of trees are genealogies and organization charts. Trees are used to
`help analyze electrical circuits and to represent the structure of mathematical
`formulas. Trees also arise naturally in many different areas of computer sci(cid:173)
`ence. For example, trees are used to organize information in database sys(cid:173)
`tems and to represent the syntactic structure of source programs in compilers.
`Chapter 5 describes applications of trees in
`the representation of data.
`Throughout this book, we shall use many different variants of trees. In this
`chapter we introduce the basic definitions and present some of the more com(cid:173)
`mon tree operations. We then describe some of the more frequently used data
`structures for trees that can be used to support these operations efficiently.
`
`3.1 Basic Terminology
`
`A tree is a collection of elements called nodes, one of which is distinguished as
`a root, along with a relation ("parenthood") that places a hierarchical struc(cid:173)
`ture on the nodes. A node, like an element of a list, can be of whatever type
`we wish. We often depict a node as a letter, a string, or a number with a cir(cid:173)
`cle around it. Formally, a tree can be defined recursively in the following
`manner.
`I. A single node by itself is a tree. This node is also the root of the tree.
`Z. Suppose n
`trees with
`roots
`is a node and T1, T,, ... , T, are
`nti n2, . . • ,nk, respectively. We can construct a new tree by making n
`In this tree n is the root and
`be the parent of nodes n 1, n2, • . . , n,.
`T1, T2 , •.. , Tk are the subtrees of the root. Nodes n 1, n2, . . . ,nk are
`called the children of node n.
`Sometimes, it is convenient to include among trees the null tree, a "tree'' with
`no nodes, which we shall represent by A.
`
`Example 3.1. Consider the table of contents of a book, as suggested by Fig.
`3. l(a). This table of contents is a tree. We can redraw it in the manner
`shown in Fig. 3.l(b). The parent-child relationship is depicted by a line.
`Trees are normally drawn top-down as in Fig. 3.l(b), with the parent above
`the child.
`the node called "Book," has
`The root,
`roots
`three subtrees with
`to
`the chapters Cl, CZ, and C3. This relationship
`is
`corresponding
`represented by the lines downward from Book to Cl, CZ, and C3. Book is
`the parent of Cl, CZ, and C3, and these three nodes are the children of Book.
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`76
`
`TREES
`
`Book
`CI
`sl.l
`s 1.2
`C2
`s2. l
`s2.l .I
`s2.J .2
`s2.2
`s2.3
`C3
`
`(a)
`
`Book
`
`C2
`
`C3
`
`Cl
`
`~·1~
`I \ I I\
`/ \
`
`sl.I
`
`sl.2
`
`s2.I Q.1
`
`s2.3
`
`Q.1.1
`
`s2.l.2
`
`(b)
`
`Fig. 3.1. A table of contents and its tree representation.
`
`The third subtree, with root C3, is a tree of a single node, while the other
`two subtrees have a nontrivial structure. For example, the subtree with root
`C2 has three subtrees, corresponding to the sections s2. J, s2.2, and s2.3; the
`last two are one-node trees, while the first has two subtrees corresponding to
`the subsections s2. l. l and s2. I. 2. o
`
`Example 3.1 is typical of one kind of data that is best represented as a
`tree. In this example, the parenthood relationship stands for containment; a
`parent node is comprised of its children, as Book is comprised of Cl, C2, and
`C3. Throughout this book we shall encounter a variety of other relationships
`that can be represented by parenthood in trees.
`If n1o n2, ••• , nk is a sequence of nodes in a tree such that n; is the
`parent of n1+ 1 for 1 ,;; i < k, then this sequence is called a path from node n 1
`to node nk. The length of a path is one less than the number of nodes in the
`path. Thus there is a path of length zero from every node to itself. For
`example, in Fig. 3.1 there is a path of length two, namely (C2, s2.J, s2.1.2)
`from C2 to s2. l.2.
`If there is a path from node a to node b, then a is an ancestor of b, and b
`is a descendant of a. For example, in Fig. 3.1, the ancestors of s2.l, are
`itself, C2, and Book, while its descendants are itself, s2.l.J, and s2.l.2.
`Notice that any node is both an ancestor and a descendant of itself.
`An ancestor or descendant of a node, other than the node itself, is called
`a proper ancestor or proper descendant, respectively. Jn a tree, the root is the
`only node with no proper ancestors. A node with no proper descendants is
`called a leaf. A subtree of a tree is a node, together with all its descendants.
`The height of a node in a tree is the length of a longest path from the
`node to a leaf. In Fig. 3.1 node Cl has height I, node C2 height 2, and node
`C3 height 0. The height of a tree is the height of the root. The depth of a
`node is the length of the unique path from the root to that node.
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`3. l BASIC TERMINOLOGY
`
`The Order of Nodes
`
`77
`
`The children of a node are usually ordered from left-to-right. Thus the two
`trees of Fig, 3.2 are different because the two chilpren of node a appear in a
`different order in the two trees. If we wish explicitly to ignore the order of
`children, we shall refer to a tree as an unordered tree.
`
`c
`
`Fig. 3.2. Two distinct (ordered) trees.
`
`The "left-to-right" ordering of siblings (children of the same node) can be
`extended to compare any two nodes that are not related by the ancestor(cid:173)
`descendant relationship. The relevant rule is that if a and b are siblings, and
`a is to the left of b, then all the descendants of a are to the left of all the des(cid:173)
`cendants of b.
`Example 3.2. Consider the tree in Fig. 3.3. Node 8 is to the right of node 2,
`to the left of nodes 9, 6, 10, 4, and 7, and neither left nor right of its ances(cid:173)
`tors I, 3, and 5.
`
`1
`
`~\~
`/~I
`I \
`I
`
`4
`
`7
`
`6
`
`10
`
`2
`
`3
`
`5
`
`8
`
`9
`
`Fig. 3.3. A tree.
`
`A simple rule, given a node n, for finding those nodes to its left and those
`to its right, is to draw the path from the root to n. All nodes branching off to
`the left of this path, and all descendants of such nodes, are to the left of n.
`All nodes and descendants of nodes branching off to the right are to the right
`of n. o
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`78
`
`TREES
`
`Preorder, Postorder, and lnorder
`
`There are several useful ways in which we can systematically order all nodes
`of a tree. The three most important orderings are called preorder, inorder
`and postorder; these orderings are defined recursively as follows.
`•
`If a tree T is null, then the empty list is the preorder, inorder and post(cid:173)
`order listing of T.
`If T consists a single node, then that node by itself is the preorder,
`in order, and postorder listing of T.
`Otherwise, let T be a tree with root n and subtrees T" T2, ... , Tb as sug(cid:173)
`gested in Fig. 3.4.
`
`•
`
`...
`
`Fig. 3.4. Tree T,
`
`L The preorder listing (or preorder traversal) of the nodes of T is the root n
`of T followed by the nodes of T1 in preorder, then the nodes of T2 in
`preorder, and so on, up to the nodes of Tk in preorder.
`2. The inorder listing of the nodes of T is the nodes of T 1 in inorder, fol(cid:173)
`lowed by node n, followed by the nodes of T 2, ••• , Tb each group of
`nodes in inorder.
`•
`3. The postorder listing of the nodes of T is the nodes of T 1 in postorder,
`then the nodes of T2 in postorder, and so on, up to Tb all followed by
`node n.
`Figure 3.5(a) shows a sketch of a procedure to list the nodes of a tree in
`preorder. To make it a postorder procedure, we simply reverse the order of
`steps (I) and (2). Figure 3.5(b) is a sketch of an inorder procedure. In each
`case, we produce the desired ordering of the tree by calling the appropriate
`procedure on the root of the tree.
`
`Example 3.3. Let us list the tree of Fig. 3.3 in preorder. We first list I and
`then call PREORDER on the first subtree of I, the subtree with root 2. This
`subtree is a single node, so we simply list it. Then we proceed to the second
`subtree of I, the tree rooted at 3. We list 3, and then call PREORDER on
`the first subtree of 3. That call results in listing 5, 8, and 9, in that order.
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`3.1 BASIC TERMINOLOGY
`
`79
`
`(I)
`(2)
`
`procedure PREORDER ( n: node);
`begin
`list n;
`for each child c of n, if any, in order from the left do
`PREORDER(c)
`end; { PREORDER }
`
`(a) PREORDER procedure.
`
`procedure !NORDER ( n: node);
`begin
`if n is a leaf then
`list n
`else begin
`INORDER(leftmost child of n);
`list n;
`for each child c of n, except for the leftmost,
`in order from the left do
`INORDER(c)
`
`end
`end; { !NORDER }
`
`(b) !NORDER procedure.
`
`Fig. 3.5. Recursive ordering procedures.
`
`Continuing in this manner, we obtain the complete preorder traversal of Fig.
`3.3: I, 2, 3, 5, 8, 9, 6, IO, 4, 7.
`Similarly, by simulating Fig. 3.5(a) with the steps reversed, we can dis(cid:173)
`cover that the postorder of Fig. 3.3 is 2, 8, 9, 5, IO, 6, 3, 7, 4, I. By simulat(cid:173)
`ing Fig. 3.5(b), we find that the inorder listing of Fig. 3.3 is 2, I, 8, 5, 9, 3,
`JO, 6, 7, 4. D
`
`A useful trick for producing the three node orderings is the following.
`Imagine we walk around the outside of the tree, starting at the root, moving
`counterclockwise, and staying as close to the tree as possible; the path we have
`in mind for Fig. 3.3 is shown in Fig. 3.6.
`For preorder, we list a node the first time we pass it. For postorder, we
`list a node the last time we pass it, as we move up to its parent. For inorder,
`we list a leaf the first time we pass it, but list an interior node the second time
`we pass it. For example, node I in Fig. 3.6 is passed the first time at the
`beginning, and the second time while passing through the "bay" between
`nodes 2 and 3. Note that the order of the leaves in the three orderings is
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`80
`
`TREES
`
`Fig. 3.6. Traversal of a tree.
`
`always the same left-to-right ordering of the leaves. It is only the ordering of
`the interior nodes and their relationship to the leaves that vary among the
`three. D
`
`Labeled Trees and Expression Trees
`
`Often it is useful to associate a label, or value, with each node of a tree, in
`the same spirit with which we associated a value with a list element in the pre(cid:173)
`vious chapter. That is, the label of a node is n11t the name of the node, but a
`value that is "stored" at the node. In some applications we shall even change
`the label of a node, while the name of a node remains the same. A useful
`analogy is tree:list = label:element = node:position.
`Example 3.4. Figure 3. 7 shows a labeled tree representing the arithmetic
`expression (a+b) * (a+c), where n 1, ••• , n7 are the names of the nodes,
`and the labels, by convention, are shown next to the nodes. The rules
`whereby a labeled tree represents an expression are as follows:
`I. Every leaf is labeled by an operand and consists of that operand alone.
`For example, node n4 represents the expression a.
`2. Every interior node n is labeled by an operator. Suppose n is labeled by a
`binary operator 6, such as + or •, and that the left child represents
`expression E 1 and the right child E2• Then n represents expression
`(E 1) 6 (E2). We may remove the parentheses if they are not necessary.
`For example, node n2 has operator +, and its left and right children
`represent the expressions a and b, respectively. Therefore, n2 represents
`(a)+(b), or just a+b. Node n1 represents (a+b)•(a+c), since• is the label
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`3.1 BASIC TERMINOLOGY
`
`81
`
`at n1o and a+b and a+c are the expressions represented by n2 and n3, respec(cid:173)
`tively. o
`
`Fig. 3.7. Expression tree with labels.
`
`Often, when we produce the preorder, inorder, or postorder listing of a
`tree, we prefer to list not the node names, but rather the labels. In the case
`of an expression tree, the preorder listing of the labels gives us what is known
`as the prefix form of an expression, where the operator precedes its left
`operand and its right operand. To be precise, the prefix expression for a sin(cid:173)
`gle operand a is a itself. The prefix expression for (E 1) 0 (E2), with 0 a
`binary operator, is 0P 1P,, where P 1 and P2 are the prefix expressions for E 1
`and E2• Note that no parentheses are necessary in the prefix expression, since
`we can scan the prefix expression 0P 1P2 and uniquely identify P 1 as the shor(cid:173)
`test (and only) prefix of P 1P 2 that is a legal prefix expression.
`For example, the preorder listing of the labels of Fig. 3.7 is •+ab+ac.
`The prefix expression for n2 , which is +ab, is the shortest legal prefix of
`+ab+ac.
`Similarly, a postorder listing of the labels of an expression tree gives us
`what is known as the postfix (or Polish) representation of an expression. The
`expression (E 1) 0 (£2) is represented by the postfix expression P 1P 26, where
`P 1 and P 2 are the postfix representations of E 1 and E 2, respectively. Again,
`no parentheses are necessary in the postfix representation, as we can deduce
`what P2 is by looking for the shortest suffix of P 1P2 that is a legal postfix
`expression. For example, the postfix expression for Fig. 3.7 is ab+ac+*. If
`we write this expression as P 1P2•, then P2 is ac+, the shortest suffix of
`ab+ac+ that is a legal postfix expression.
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`82
`
`TREES
`
`The inorder traversal of an expression tree gives the infix expression
`itself, but with no parentheses. For example, the inorder listing of the labels
`of Fig. 3. 7 is a+ b • a+ c. The reader is invited to provide an algorithm for
`traversing an expression tree and producing an infix expression with all
`needed pairs of parentheses.
`
`Computing Ancestral Information
`
`The preorder and postorder traversals of a tree are useful in obtaining ances(cid:173)
`tral information. Suppose postorder(n) is the position of node n in a post(cid:173)
`order listing of the nodes of a tree. Suppose desc(n) is the number of proper
`descendants of node n. For example, in the tree of Fig. 3.7 the postorder
`numbers of nodes n2 , n4 , and n5 are 3, 1, and 2, respectively.
`The postorder numbers assigned to the nodes have the useful property
`that the nodes in the subtree with root n are numbered consecutively from
`postorder(n) - desc(n) to postorder(n). To test if a vertex x is a descendant
`of vertex y, all we need do is determine whether
`
`postorder(y)-desc(y) ,,;; postorder(x) ,,;; postorder(y).
`
`A similar property holds for preorder.
`
`3.2 The ADT TREE
`
`In Chapter 2, lists, stacks, queues, and mappings were treated as abstract data
`types (ADT's). In this chapter trees will be treated both as ADT's and as
`data structures. One of our most important uses of trees occurs in the design
`of implementations for the various ADT's we study. For example, in Section
`5.1, we shall see how a "binary search tree" can be used to implement
`abstract data types based on the mathematical model of a set, together with
`operations such as INSERT, DELETE, and MEMBER (to test whether an
`element is in a set). The next two chapters present a number of other tree
`implementations of various ADT's.
`In this section, we shall present several useful operations on trees and
`show how tree algorithms can be designed in terms of these operations. As
`with lists, there are a great variety of operations that can be performed on
`trees. Here, we shall consider the following operations:
`I. PARENT(n, T). This function returns the parent of node n in tree T. If
`n is the root, which has no parent, A is returned. In this context, A is a
`"null node," which is used as a signal that we have navigated off the tree.
`2. LEFTMOST _CHILD(n, T) returns the leftmost child of node n in tree T,
`and it returns A if n is a leaf, which therefore has no children.
`3. RIGHT_SIBLING(n, T) returns the right sibling of node n in tree T,
`defined to be that node m with the same parent p as n such that m lies
`immediately to the right of n in the ordering of the children of p. For
`in Fig. 3.7, LEFTMOST_CHILD(n 2) = n4;
`example,
`for
`the
`tree
`RIGHT_SIBLING(n4) = n5, and RIGHT_SIBLING (ns) = A.
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`3.2 THE ADT TREE
`
`83
`
`4. LABEL(n, T) returns the label of node n in tree T. We do not, however,
`require labels to be defined for every tree.
`5. CREATEi(v, T 1• Tz, ... , T;) is one of an infinite family of functions,
`one for each value of i = 0, I, 2, . . .. CREATEi makes a new node r
`label v and gives it i children, which are the roots of trees
`with
`Ti. T2 , . . . , T;, in order from the left. The tree with root r is returned.
`Note that if i = 0, then r is both a leaf and the root.
`6. ROOT(T) returns the node that is the root of tree T, or A if T is the null
`tree.
`7. MAKENULL(T) makes T be the null tree.
`
`Example 3.5. Let us write both recursive and nonrecursive procedures to take
`a tree and list the labels of its nodes in preorder. We assume that there are
`data types node and TREE already defined for us, and that the data type
`TREE is for trees with labels of the type labeltype. Figure 3.8 shows a recur(cid:173)
`sive procedure that, given node n, lists the labels of the subtree rooted at n in
`preorder. We call PREORDER(ROOT(T)) to get a preorder listing of tree T.
`
`p·rocedure PREORDER ( n: node );
`{ list the labels of the descendants of n in preorder }
`var
`
`c: node;
`begin
`print(LABEL(n, T));
`c := LEFTMOST_CHILD(n, T);
`while c < > A do begin
`PREORDER(c);
`c := RIGHT_SIBLING(c, T)
`
`end
`end; { PREORDER }
`
`Fig. 3.8. A recursive preorder listing procedure.
`
`We shall also develop a nonrecursive procedure to print a tree in
`preorder. To find our way around the tree, we shall use a stack S, whose
`type STACK is really "stack of nodes." The basic idea underlying our algo(cid:173)
`rithm is that when we are at a node n, the stack will hold the path from the
`root to n, with the root at the bottom of the stack and node n at the top. t
`
`t Recall our discussion of recursion in Section 2.6 in which we illustrated how the implementation
`If we examine Fig. 3.8, we can
`of a recursive procedure involves a stack of activation records.
`observe that when PREORDER(n) is called, lhe active procedure calls, and therefore the stack of
`activation records, correspond to the calls of PREORDER for all the ancestors of n. Thus our
`nonrecursive preorder procedure, like the ex.ample in Section 2.6, models closely the way the re~
`cursive procedure is implemented.
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`84
`
`TREES
`
`One way to perform a nonrecursive preorder traversal of a tree is given
`by the program NPREORDER shown in Fig. 3.9. This program has two
`modes of operation. In the first mode it descends down the leftmost unex(cid:173)
`plored path in the tree; printing and stacking the nodes along the path, until it
`reaches a leaf.
`The program then enters the second mode of operation in which it retreats
`back up the stacked path, popping the nodes of the path off the stack, until it
`encounters a node on the path with a right sibling. The program then reverts
`back to the first mode of operation, starting the descent from that unexplored
`right sibling.
`The program begins in mode one at the root and terminates when the
`stack becomes empty. The complete program is shown in Fig. 3.9.
`
`3.3 Implementations of Trees
`
`In this section we shall present several basic implementations for trees and dis(cid:173)
`cuss their capabilities for supporting the various tree operations introduced in
`Section 3.2.
`
`An Array Representation of Trees
`
`Let T be a tree in which the nodes are named J, 2, ... , n. Perhaps the sim(cid:173)
`plest representation of T that supports the PARENT operation is a linear
`array A in which entry A [i] is a pointer or a cursor to the parent of node i.
`The root of T can be distinguished by giving it a null pointer or a pointer to
`itself as parent. In Pascal, pointers to array elements are not feasible, so we
`shall have to use a cursor scheme where A[i] = j if node j is the parent of
`node i, and A[i] = 0 if node i is the root.
`This representation uses the property of trees that each node has a unique
`parent. With this representation the parent of a node can be found in con(cid:173)
`stant time. A path going up the tree, that is, from node to parent to parent,
`and so on, can be traversed in time proportional to the number of nodes on
`the path. We can also support the LABEL operator by adding another array
`L, such that L(i] is the label of node i, or by making the elements of array A
`be records consisting of an integer (cursor) and a label.
`
`Example 3.6. The tree of Fig. 3. lO(a) has the parent representation given by
`the array A shown in Fig. 3. IO(b). o
`
`The parent pointer representation does not facilitate operations that
`require child-of information. Given a node n, it is expensive to determine the
`children of n, or the height of n.
`In addition, the parent pointer representa(cid:173)
`tion does not specify the order of the children of a node. Thus, operations
`like LEFTMOST_CHILD and RIGHT_SIBLING are not well defined. We
`could impose an artificial order, for example, by numbering the children of
`each node after numbering the parent, and numbering the children in
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`3.3 IMPLEMENT A T!ONS OF TREES
`
`85
`
`procedure NPREORDER ( T: TREE );
`{ nonrecursive preorder traversal of tree T }
`
`var
`
`m: node; { a temporary }
`S: STACK; { stack of nodes holding path from the root
`to the parent TOP(S) of the "current" node m }
`
`begin
`{ initialize }
`MAKENULL(S);
`m := ROOT(T);
`
`while true do
`if m < > A then begin
`print(LABEL(m, T));
`PUSH(m, S);
`{ explore leftmost child of m }
`m := LEFTMOST_CHILD(m, T)
`end
`else begin
`{ exploration of path on stack
`is now complete }
`if EMPTY(S) then
`return;
`{ explore right sibling of node
`on top of stack }
`m := RIGHT_SIBLING(TOP(S), T);
`POP(S)
`end
`end; { NPREORDER }
`
`Fig. 3.9. A nonrecursive preorder procedure.
`
`increasing order from left to right. On that assumption, we have written the
`function RIGHT_SIBLING in Fig. 3.11, for types node and TREE that are
`defined as follows:
`
`type
`node = integer;
`TREE = array [l..maxnodes] of node;
`
`For this implementation we assume the null node A is represented by 0.
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`86
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`TREES
`
`(a} a tree
`
`10
`9
`8
`7
`6
`5
`4
`3
`2
`1
`AO I 2 2 5 5 5 3 3
`
`(b) parent representation.
`
`Fig. 3.10. A tree and its parent pointer representation.
`
`function RIGHT_SIBLING ( n: node; T: TREE) : node;
`{ return the right sibling of node n in tree T }
`•var
`
`i, parent: node;
`begin
`parent:= T[n];
`for i : = n + 1 to maxnodes do
`{ search for node after n with same parent }
`if T[i) = parent then
`return (i);
`return (0) { null node will be returned
`if no right sibling is ever found }
`end; { RIGHT_SIBLING}
`
`Fig. 3.11. Right sibling operation using array representation.
`
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`3.3 IMPLEMENTATIONS OF TREES
`
`87
`
`Representation of Trees by Lists of Children
`
`An important and useful way of representing trees is to form for each node a
`list of its children. The lists can be represented by any of the methods sug(cid:173)
`gested in Chapter 2, but because the number of children each node may have
`can be variable, the linked-list representations are often more appropriate.
`Figure 3.12 suggests how the tree of Fig. 3.lO(a) might be represented.
`There is an array of header cells, indexed by nodes, which we assume to be
`numbered l, 2, ... , 10. Each header points to a linked list of "elements,"
`which are nodes. The elements on the list headed by header[i] are the chil(cid:173)
`dren of node i; for example, 9 and 10 are the children of 3.
`
`4
`
`6
`
`7
`
`=r 1 3 ] . l
`-•
`_r 2 J
`-l -
`_r 1 . l
`J
`I
`::r
`-l 5
`-r JO ] . l
`9 J
`- t
`I
`::r
`1.
`t 1 8 1 • J
`J J 1 1
`
`2
`3
`4
`5
`6
`7
`8
`9
`JO
`
`• •
`•
`•
`•
`•
`
`header
`
`Fig. 3.12. A linked-list representation of a tree.
`
`Let u