Uieait)
`AND ALGORITHMS
`
`ALFRED V. AHO
`JOHN E. HOPCROFT
`
`Maal ome MeF
`
`4
`
`EX 1010
`
`1
`
`EX 1010
`
`

`

`
`
`
`
`
`Data Structures
`
`and Algorithms
`
`ALFRED V. AHO
`
`Bell Laboratories
`Murray Hill, New Jersey
`
`JOHN E. HOPCROFT
`
`Corneil University
`ithaca, New York
`
`JEFFREY BD. ULLMAN
`
`
`
`
`
`
`
`
`
`
`
`Stanford University
`Stanford, California
`
`
`
`VV.ADDISON-WESLEYPUBLISHINGCOMPANY.
`Reading, Massachusetts.
`lo Park, California
`ondon®Amsterdam@ Don Mills, Ontario.@ Sydney’
`
`
`2
`
`

`

`
`
`This book is in the
`ADDISON-WESLEY SERIES IN
`COMPUTER SCIENCE AND INFORMATION PROCESSING
`
`Michael A. Harrison
`Consulting Editor
`
`
`
`
`
`
`
`
`
`Library of Congress Cataloging in Publication Data
`Aho, Alfred V.
`Data structures and algorithms.
`2. Algorithms.
`1. Data structures (Computer science)
`L. Hopcroft, John E., 1939-
`.
`HH, Ullman,
`Jeffrey D., 1942-
`.
`IIL. Title.
`QA76.9.D35A38
`1982
`001.64
`82-11596
`ISBN 0-201-00023-7
`
`
`
`
`
`Reproduced by Addison-Wesley from camera-ready copy supplied by the authors.
`
`Reprinted with corrections April, 1987
`Copyright © 1983 by Bell Telephone Laboratories, Incorporated.
`
`All rights reserved. No part of this publication may be reproduced, stored in a re-
`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-
`
`
`
`
`
`lisher. Printed in the United States of America. Published simultaneously in Canada. 2 ISBN: 0-201-00023-7
`
`3
`
`

`

`Contents
`
`Design and Analysis of Algorithms
`From Problems to Programs ...........:---s-ceensereneeneeetnrn ee rer ee neees 1
`Abstract Data Types ......0:ccccceesseeeeeeeerene nearer nee tne ee 10
`Data Types, Data Structures, and Abstract Data Types... cece 13
`The Running Time of a Program ......cseecccceereeetreeeeereere eters 16
`Calculating the Running Time of a Program.......:::-sceeeeeeees 21
`Good Programming Practice .........:.ccsceeceereee sees renee tener ents27
`Super Pascal......sccccccecccececereeetesteeeeenneeseereeeeee sete e eres sees29
`
`Chapter 1
`
`1 1
`
`.2
`1.3
`L4
`1.5
`1.6
`1.7
`
`Chapter 2
`2.1
`2.2
`2.3
`2.4
`2.3
`2.6
`
`Basic Data Types
`The Data Type “List” ......ccceccccsessereetenetnsrecnnn ene ceeeaneneese eee37
`Implementation of Lists ........:.0..:cccecerieeeerstertennenestrttees40
`Stack..ccccccecucceceeea rene et een eee ee epee nEER EGTA EERE STEER ERE EASE EGET SS 33
`QUGUES oe e eee enr eeEEE56
`Mappings .......cc-ccceeeeeeeeeeereeeee cnn ernenesneaee nen esuoueeeeneeneeeees61
`Stacks and Recursive Procedures ........:-::eceeseeseeeeeerenseeree tens64
`
`Chapter 3
`3.1
`3.2
`3.3
`3.4
`
`Trees
`Basic Terminology ............::ceereeseeeeeceeee rere een ree75
`The ADT TREE......cccccece ere ee ee eee ee re en EE Ee EERE EEE 82
`Implementations Of Trees .........-cscerse reece erent tate e terete84
`Binary Trees ...0.ccseeee see ce etree reer terre ree eeer ener teeter regs93
`
`Chapter 4
`4.1
`4.2
`4.3
`4.4
`4.5
`4.6
`4.7
`4.8
`4.9
`4.10
`4.11
`4,12
`
`Basic Operations on Sets
`Introduction to Sets .....:..c:cccececet erent eee nen eee e eens ened enersneesntas 107
`An ADT with Union, Intersection, and Difference ............-+-- 109
`A Bit-Vector Implementation of Sets............::..ceeeeeeeenrer cree 112
`A Linked-List Implementation of Sets ............:::: see scneeeeeeees 115
`The Dictionary ........-.:ccceeeeceeeeece rece teen esr ee eee e reer eres 117
`Simple Dictionary Implementations.......-..--.eseeeereeeetee steerer 119
`The Hash Table Data Structure.......:ces-ccceseeee eee ee eter e rennet es 122
`Estimating the Efficiency of Hash Functions..........-- ee 129
`Implementation of the Mapping ADT ...........::::eceeeeeees tener ees 135
`Priority QUeUeS .........0ccce eter teerrtrete ee eeeer reenter ert e rene ri 135
`Implementations of Priority Queues ..........: seer reer 138
`Some Complex Set Structures ........0.. creer teeter etter 145
`
`4
`
`

`

`CONTENTS
`
`Advanced Set Representation Methods
`
`Binary Search Trees ...2.......:...ccssescteeeseeteceeeteeeaeneeetteteee 155
`Time Analysis of Binary Search Tree Operations .................. 160
`TYICS oe eceecc eee cnee eee eee n est eee ee teaee ner teneegeteenepeenecene eenseges 163
`Balanced Tree Implementations of Sets ............0..cc:cceneeenseoes 169
`Sets with the MERGE and FIND Operations.............,-......5 180
`An ADT with MERGE and SPLIT .................ceeeceeeeneeeneenee 189
`
`Directed Graphs
`
`Basic Defimitions .............ccccseceereneeeeetareeneeeaeeeeenesseebetenes 198
`Representations for Directed Graphs..........ccccccesceseeeeenneeenes 199
`The Single-Source Shortest Paths Problem ............0.:0eceeeee203
`The All-Pairs Shortest Path Problem .....0.........ccceeeeeeeereeerees 208
`Traversals of Directed Graphs .............0.ccccccscceeeeeeseneeeeeraes 215
`Directed Acyclic Graphs.......cccccccsctsecetsteetiieeetieeeereeeies 219
`Strong Components .....0..0 ccc ceecseecetesetee een etenteseetaneen tenes 222
`
`Undirected Graphs
`Definitions ............ 00...eekeeebeetaeeeeeeteeteeeseeeeseetaarsneesecees 230
`Minimum-Cost Spanning Trees ........00..00.. cee ceeceseeee sere enene 233
`TYAVETSA]S 00. ...ececece ee etee renee erento eter estan ean REAR Eee rE entEEE 239
`Articulation Points and Biconnected Components.............0.0.244
`Graph Matching ............cccccceseceeeeeteeetse eens renpeneteneeeeaneeeres 246
`
`Local Search Algorithms .............0::cccccseeteeseree ersten eeeeennines 336
`
`Sorting
`
`The Internal Sorting Model,.........:c-.::seseeeeceeeesersensee een ganees253
`Some Simple Sorting Schemes .............::cuesciceeseueresreeeteeenes 254
`QUICKSOLE 0. cece cece t ceed nent ec been band teed ereda nt baed een eeetos 260
`Heapsort 0... .... cece ceeeecceet acne eeeeneesseeaseeeesneesseeesseneeeaeeees27k
`Bin Sorting .......0..cccecceesee eee ceetensnneeen tree seeeteeneneeeantentnanenes274
`A Lower Bound for Sorting by Comparisons...............-::.50008 282
`Order Statistics............ 0c ccseeessseseeceesaeeseestaeeseeeten ten enatenes 286
`
`Algorithm Analysis Techniques
`
`Efficiency of Algorithms ...............ccccisces eee seeeeeneeeeeee ere ene ne 293
`Analysis of Recursive Programs ....,.....:0.:cccseseeeseeeeaeneeesneees294
`Solving Recurrence Equations .........:::ccscccsceressueetereesteunaees296
`A General Solution for a Large Class of Recurrences ............ 298
`
`Algorithm Design Techniques
`
`Divide-and-Conquer Algorithms...............c:esccccseseesceeeeseeeee306
`Dynamic Programming ......-..ccececcareseeeeeteeenreeeeseeeenrenapenies 311
`Greedy Algorithms ...........ceccereeeeeceeeetesereerreeueteeatieeeeseas 321
`Backtr acking .........cc.ccccccc sete senses cere ee ee tae etn e eee emen een eearegs324
`
`5
`
`Chapter 5
`3.1
`5.2
`3.3
`5.4
`5.5
`3.6
`
`Chapter 6
`6.1
`6.2
`6,3
`6.4
`6.5
`6.6
`6.7
`
`Chapter 7
`7.1
`72
`73
`74
`7.5
`
`Chapter 8
`8.1
`8.2
`8.3
`8.4
`8.5
`8.6
`8.7
`
`Chapter 9
`9.1
`9.2
`93
`9.4
`
`Chapter 10
`10.1
`10.2
`10.3
`10.4
`10.5
`
`
`
`

`

` xi
`CONTENTS
`
`Chapter 11
`il.
`11.2
`11.3
`11.4
`
`Data Structures and Algorithms for External Storage
`A Mode! of External Computation........-c-ecrersseriseresrtsertes347
`External Sorting .....cccceceesccrsseeee setteeerrr349
`Storing Information in Files ......6:-s-:csecssereereetereer ee361
`External Search Trees .....cccceecectrrere terete368
`
`Chapter 12
`12.5
`12.2
`12.3
`12.4
`12.5
`12.6
`
`Memory Management
`The Issues in Memory Management..........sssesrrecrrsrrerersee378
`Managing Equal-Sized BIOCKS ...cc.cceceeeennee ee ree ett ence eee nersees382
`Garbage Collection Algorithms for Equal-Sized Blocks ........--384
`Storage Allocation for Objects with Mixed Sizes 0.0... cere renee ees392
`Buddy Systems .......c2-cccscecerseseees presenceree400
`Storage Compaction .......-cccectessrrterteneeeese404
`Bibliography ........-:cccccceceee seerseee4il
`TAGOX 5c cccccceccs cence tere een EEEET419
`
`
`6
`
`
`
`

`

`
`
`
`
`
`
`
`
`
`
`
`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-
`ence. For example,
`trees are used to organize information in database sys-
`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-
`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-
`ture on the nodes. A node, like an elementofa list, can be of whatever type
`we wish, We often depict a node as a letter, a string, or a number with a cir-
`cle around it. Formally, a tree can be defined recursively in the following
`manner,
`
`1, A single node byitself is a tree. This node is also the root of the tree.
`2.
`Suppose
`a
`is
`a
`node
`and
`17 ,,7>,...,7,
`are
`trees with
`roots
`ny, 2, ... Mg, Tespectively. We can construct a new tree by making n
`be the parent of nodes mj, m2,...,%%.
`In this tree # is the root and
`T,,7T2,...,7, are the subtrees of the root. Nodes my, m2, ...,M, are
`called the children of node x.
`
`
`
`
`
`
`
`
`
`
`Sometimes, it is convenient to include among trees the null tree, a “tree”? with
`no nodes, which we shail represent by A.
`
`Example 3.1. Consider the table of contents of a book, as suggested by Fig.
`-3.1({a). This table of contents is a tree, We can redraw it
`in the manner
`
`Shown in Fig. 3.1(b). The parent-child relationship is depicted by a line.
`
`‘Trees are normally drawn top-down as in Fig. 3.1(b), with the parent above
`‘thechild.
`
`three subtrees with roots
`the node called “Book,” has
`:
`:
`“The root,
`COtresponding to the chapters Cl, C2,
`and C3.
`This
`relationship is
`
`represented by the lines downward from Book to Cl, C2, and C3. Book is
`he parent of C1, C2, and C3, and these three nodes are the children of Book.
`
`
`
`7
`
`
`
`

`

`76
`
`Book
`Ci
`sl.
`
`C22
`52.2
`
`$2.1.1
`82.4.2
`
`$2.3
`C3
`
`(a)
`
`TREES
`
`C3
`
`.
`
`cl
`
`Book
`
`C2
`
`sl.
`
`sl.2
`
` s2.1
`
`82.2
`
`82.3
`
`oo
`
`JN SIN
`/ \
`
`s2.t.4
`
`82.1.2
`
`(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 s2.1, 52.2, and s2.3; the
`last two are one-node trees, while the first has two subtrees corresponding to
`the subsections 2.1.1 and s2.1.2. 0
`
`'
`
`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 C1, C2, and
`C3. Throughout this book we shall encounter a variety of other relationships
`that can be represented by parenthoodin trees.
`is the
`If my, m2, ...,My iS @ Sequence of nodes in a tree such that n,;
`parent of n;,; for 1 = i < k, then this sequence is called a path from node nj,
`to node n,. 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
`exampie,
`in Fig. 3.1 there is a path of length two, namely (C2, 52.1, 2.1.2)
`from C2 to s2.1.2.
`If there is a path from node a io 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 s2.1, are
`itself, C2, and Book, while its descendants.are itself, 82.1.1, and 2.1.2.
`Notice that any node is both an ancestor and a descendant ofitself.
`An ancestor or descendant of a node, other than the nodeitself, is called
`a proper ancestor or proper descendant, respectively.
`In 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. 3.4 node C1 has height 1, 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 nade,
`
`8
`
`
`
`

`

`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 children of node a appear in a
`different order in the two trees. [f we wish explicitly to ignore the order of
`children, we shall refer to a tree as an unordered tree.
`
`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-
`descendant relationship. The relevant rule is that if a and b are siblings, and
`a is to the left of 6, then all the descendants of a are to the left ofall the des-
`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-
`tors 1, 3, and 5.
`
`The Order of Nodes
`
` 3.1 BASIC TERMINOLOGY
`
`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 tom. AH nodes branching off to
`the left of this path, and ali descendants of such nodes, are to the left of a.
`: All nodes and descendants of nodes branching off to the right are to the right
`‘of n.o
`
`a ™N
`\
`|
`
`10
`
`8
`
`9
`
`Fig. 3.3. A tree.
`
`9
`
`
`
`

`

`Preorder, Postorder, and [norder
`
`TREES
`
`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 atree T is null, then the empty list is the preorder, inorder and post-
`order listing of 7.
`then that node by itself is the preorder,
`If T consists a single node,
`inorder, and postorderlisting of T.
`Otherwise, let T be a tree with root mn and subtrees 7), 7), ...,7,, as sug-
`gested in Fig. 3.4,
`
`@
`
` 78
`
`Fig. 3.4, Tree T.
`
`1. The preorder listing (or preorder traversal) of the nodes of T is the root x”
`of T followed by the nodes of T,
`in preorder, then the nodes of T,
`in
`preorder, and so on, up to the nodes of 7, in preorder.
`in inorder, fol-
`2. The inorder listing of the nodes of T is the nodes of 7,
`lowed by node n, followed by the nodes of T2,...,7,, each group of
`nodes in inorder.
`.
`3. The postorder listing of the nodes of T is the nodes of T,
`in postorder,
`then the nodes of T,
`in postorder, and so on, up to 7;,, 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 (1) 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 L and
`then call PREORDER onthefirst subtree of 1, the subtree with root 2. This
`subtree is a single node, so we simply list it. Then we proceed to the second
`subtree of 1, 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.
`
`10
`10
`
`
`
`

`

`
`
`
`
`
`
`
`
`
`
`
`
`
`3.1 BASIC TERMINOLOGY
`
`79
`
`(1)
`(2)
`
`procedure PREORDER( m: node );
`begin
`list 1;
`for each child c of n, if any, in order from the left do
`PREORDER(c)
`{ PREORDER}
`
`end;
`
`(a) PREORDERprocedure.
`
`procedure INORDER( n: node );
`begin
`if n is a leaf then
`list
`else begin
`INORDER (leftmost child of 7);
`list n;
`for each child c of n, except for the leftmost,
`in order from the left do
`INORDER(c)
`
`end
`{ INORDER}
`
`end;
`
`(b) INORDER procedure.
`
`Fig. 3.5. Recursive ordering procedures.
`
`
`Continuing in this manner, we obtain the complete preorder traversal of Fig.
`3.3: 1, 2, 3, 5, 8, 9, 6, 10, 4, 7.
`
`Similarly, by simulating Fig, 3.5(a) with the steps reversed, we can dis-
`-" caver that the postorder of Fig, 3.3 is 2, 8, 9, 5, 10, 6, 3, 7, 4, 1. By simulat-
`
`.ing Fig. 3.5(b), we find that the inorder listing of Fig. 3.3 is 2, 1, 8, 5, 9, 3,
`
`10, 6,7, 4.0
`
`
`
`
`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
`“cst a node the last time we pass it, as we move up to its parent. For inorder,
`welist a leaf the first time we pass it, but list an interior node the second time
`wepass it. For example, node 1
`in Fig. 3.6 is passed the first time at the
`beginning, and the second time while passing through the “bay” between
`‘Modes 2 and 3. Note that
`the order of the leaves in the three orderings is
`
`
`
`11
`11
`
`
`
`ahceeeis
`
`
`

`

`
`
`
`
`TREES
`
`fg f\9 ]
`
`10
`
`Fig. 3.6. Traversal of a tree.
`
`it is only the ordering of
`always the sameleft-to-right ordering of the leaves.
`the interior nodes and their relationship to the leaves that vary among the
`three,
`
`Labeled Trees and Expression Trees
`
` 80
`
`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-
`vious chapter. That is, the label of a node is not 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,,...,7 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:
`1. Every leaf is labeled by an operand and consists of that operand alone.
`For example, node n, 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, and the right child £,. Then nm
`represents expression
`(E,) 0 (E,). We may remove the parentheses if they are not necessary.
`For example, node nm) has operator +, and its left and right children
`represent
`the expressions a and b, respectively, Therefore, n, represents
`(a}+(5), or just a+b... Node n, represents (a+b)*(at+c), since * is the label
`
`12
`12
`
`

`

`
`3.1 BASIC TERMINOLOGY
`81
`
`at my, and a+b and a+c are the expressions represented by nz and 13, respec-
`tively, 0
`
`
`
`Fig. 3.7. Expression tree with labels.
`
`
`inorder, or postorder listing of a
`Often, when we produce the preorder,
`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 andits right operand. To be precise, the prefix expression for a sin-
`gle operand a is a@
`itself. The prefix expression for (E,) @ (£2), with 0 a
`binary operator,
`is 0P,;P2, where P, and P2 are the prefix expressions for FE;
`and £3. Note that no parentheses are necessary in the prefix expression, since
`we can scan the prefix expression 8P,P, and uniquely identify P, as the shor-
`test (and only) prefix of P,P, 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 m2, which is +a@b,
`is the shortest legal prefix of
`+abtac,
`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,) is represented by the postfix expression P,?,0, where
`P, and P» are the postfix representations of E, and E, respectively, Again,
`no parentheses are necessary in the postfix representation, as we can deduce
`what P, is by looking for the shortest suffix of P\P, that is a legal postfix
`expression. For example, the postfix expression for Fig. 3.7 is abtact+*.
`If
`we write this expression as P\P2*,
`then P, is ac+,
`the shortest suffix of
`ab+ac+ that is a legal postfix expression.
`
`
`
`
`13
`13
`
`

`

`TREES
`
`The inorder traversal of an expression tree gives the infix expression
`itself, but with no parentheses. For exampie, the inorder listing of the labels
`of Fig. 3.7 is atb *.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
`
` 82
`
`The preorder and postorder traversals of a tree are useful in obtaining ances-
`tral information. Suppose postorder(n) is the position of node n in a post-
`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 2, ng, and ms 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 nm 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:
`If
`1, PARENT(x, 7). This function returns the parent of node # in tree T.
`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 m is a leaf, which therefore has no children.
`3. RIGHT_SIBLING(n, T)
`returns the right sibling of nede 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
`example,
`for
`the
`tree
`in Fig.
`3.7, LEFTMOST_CHILD(n2) = ny;
`RIGHT_SIBLING(n,) = ns, and RIGHT_SIBLING (ns) = A.
`
`14
`14
`
`
`
`
`
`
`
`

`

`
`
`
`
`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.
`§. CREATEI(¥, T,, Tz, ...,7,;) is one of an infinite family of functions,
`one for each value of 7 = 0, 1,2, .... CREATE: makes a new node r
`with label
`vy and gives
`it
`i children, which are the roots of
`trees
`T,, Tz, ...,7;, in order from the left. The tree with root r is returned,
`Note that if 7 = 0, then r is both a Jeaf 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 7 bethe 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-
`sive procedure that, given node n, lists the labels of the subtree rooted at ” in
`preorder. We call PREORDER(ROOT(T)) to get a preorderlisting of tree T.
`
`procedure PREORDER( a: node );
`{ list the labeis 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.
`
`tree in
`a
`We shail also develop a nonrecursive procedure to print
`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-
`rithm is that when we are at a node a, the stack will hold the path from the
`root ton, 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
`of a recursive procedure involves a stack of activation records.
`If we examine Fig, 3.8, we can
`observe that when PREORDER(») is called, the active procedure calls, and therefore the stack of
`activation records, correspond to the calls of PREORDERfor ail the ancestors of n. Thus our
`honrecursive preorder procedure,
`like the example in Section 2.6, models closely the way the re-
`cursive procedure is implemented.
`
`
`
`
`
`15
`
`

`

`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-
`plored path in the tree, printing and stacking the nodes along the path, until it
`reachesa 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-
`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 |, 2,.-.,. Perhaps the sim-
`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 Afi] = j if node j is the parent of
`node i, and A[f] = 0 if node ¢ 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-
`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[/] is the label of node f, 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. 3.10(a) has the parent representation given by
`the array A shown in Fig. 3.10(b). 0
`
`facilitate operations that
`representation does not
`The parent pointer
`require child-of information. Given a node a, it is expensive to determine the
`children of n, or the height of n.
`In addition, the parent pointer representa-
`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
`
`16
`16
`
`
`
`
`
`

`

`3.3 IMPLEMENTATIONS OF TREES
`
`85
`
`procedure NPREORDER( T: TREE );
`{ nonrecursive preorder traversal of tree T }
`var
`
`{ a temporary }
`m: node;
`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(7);
`
`while true do
`ifm <> A then begin
`print(LABEL(m, T));
`PUSH(m, 5);
`{ 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 nede
`on top of stack }
`m:= RIGHT_SIBLING(TOP(S), 7):
`POP(S)
`
`end
`{ NPREORDER}
`
`end;
`
`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. TREEthat are
`>:
`‘.. defined as follows:
`
`type
`
`node = integer;
`TREE=array [1..maxnodes] of node;
`
`
`
`
`
`
`_ Por this implementation we assume the null node A is represented by @.
`
`17
`17
`
`

`

` &6
`
`
`
`
`
`
`
`
`
`2 3
`4
`5
`6
`7
`8
`9
`10
`afolil:[2]2[5[51[s51]313|
`
`TREES
`
`iN
`/\,/%
`JN
`
`(a) a tree
`
`(b) parent representation.
`
`Fig. 3.10. A tree and its parent pointer representation.
`
`fonction RIGHTSIBLING ( n: node; 7: 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 (7);
`return (0)
`{ null node will be returned
`if no right sibling is ever found }
`{ RIGHT_SIBLING }
`
`end;
`
`Fig. 3.11. Right sibling operation using atray representation.
`
`18
`18
`
`

`

`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-
`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. 3.10(a) might be represented.
`There is an array of header cells,
`indexed by nodes, which we assume to be
`numbered 1, 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-
`dren of node i; for example, 9 and 10 are the children of 3.
`
`
`
`header
`
`Fig. 3.12. A linked-list representation of a tree.
`
`Let us first develop the data structures we need in terms of an abstract
`data type LIST (of nodes), and then give a particular implementation of lists
`and see how the abstractions fit together. Later, we shall see some of the
`simplifications we can make, We begin with the following type declarations:
`
`
`
`
`
`
`
`type
`It
`node=integer;
`LIST={ appropriate definition for list of nodes };
`position = { appropriate definition for positions in lists },
`TREE = record
`header: array [1..maxnodes] of LIST;
`labels: array [1..maxnodes] of labeltype;
`root: node
`
`end;
`
`19
`19
`
`

`

`We assume that the root of each tree is stored explicitly in the root field.
`Also, 0 is used to represent the null node.
`Figure 3.13 shows the code for the LEFTMOST_CHILD operation. The
`reader should write the code for the other operations as exercises.
`
`TREES
`
`function LEFFMOST_CHILD ( n: node; T: TREE) : node;
`{ returns the teftmost child of node n of tree T }
`var
`
`L: LIST;
`
`{ shorthand for the list of a's children }
`
`L := T.header{n |;
`if EMPTY(L) then {17 is a leaf}
`return (0}
`
`else
`
`return (RETRIEVE(FIRST(L), L)}
`{ LEFTMOST_CHILD }
`
`end;
`
`Fig. 3.13. Function to find leftmost child.
`
` 88
`
`Now let us choose a particular implementation of lists, in which both LIST
`and position are integers, used as cursors into an array celispace of records:
`var
`
`celispace: array [1..maxnodes ] of record
`nade : integer;
`next: integer
`
`end;
`
`lists of children have header cells.
`that
`insist
`To simplify, we shall not
`Rather, we shall let 7.header[n] point directly to the first cell of the list, as is.
`suggested
`by
`Fig.
`3.12.
`Figure
`3.14(a)
`shows
`the
`function
`LEFTMOST_CHILD of Fig. 3.13 rewritten for this specific implementation.
`Figure 3.14(b) shows the operator PARENT, which is more difficult to write
`using this representation of lists, since a search ofall lists is required to deter-
`mine on which list a given node appears.
`
`The Leftmost-Child, Right-Sibling Representation
`
`The data structure described above has, among other shortcomings, the inabil-
`ity to create large trees from smaller ones, using the CREATE/ op