ECE250: Algorithms and Data Structures

Trees

Materials from CLRS: Chapter 10.4 and Appendix B.5 Weiss: Chapter 4

Ladan Tahvildari, PEng, SMIEEE Professor

Software Technologies Applied Research (STAR) Group

Dept. of Elect. & Comp. Eng.

University of Waterloo

Acknowledgements

v The following resources have been used to prepare materials for this course: Ø  MIT OpenCourseWare Ø  Introduction To Algorithms (CLRS Book) Ø  Data Structures and Algorithm Analysis in C++ (M. Wiess) Ø  Data Structures and Algorithms in C++ (M. Goodrich)

v Thanks to many people for pointing out mistakes, providing suggestions, or helping to improve the quality of this course over the last ten years: Ø  http://www.stargroup.uwaterloo.ca/~ece250/acknowledgment/

Lecture 10 ECE250 2

Lecture 10 ECE250 3

New Data Structure

v Linear access time of linked lists is prohibitive Ø Does there exist any simple data structure for

which the running time of most operations (search, insert, delete) is O(log N)?

Trees

v A tree is a collection of nodes Ø  The collection can be empty Ø  (recursive definition) If not empty, a tree consists of a

distinguished node r (the root), and zero or more nonempty subtrees T1, T2, ...., Tk, each of whose roots are connected by a directed edge from r

Lecture 10 ECE250 4

Some Terminologies v  Parent and Child

Ø  Every node except the root has one parent Ø  A node can have an arbitrary number of children

v  Leaves Ø  Nodes with no children

v  Sibling Ø  Nodes with same parent

Lecture 10 ECE250 5

Some Terminologies v Path

Ø  A sequence of nodes such that is the parent of for

v Length Ø  Number of edges on the path

v Depth of a node Ø  Length of the unique path from the root to that node Ø  The depth of a tree is equal to the depth of the deepest leaf

v Height of a node Ø  Length of the longest path from that node to a leaf Ø  All leaves are at height 0 Ø  The height of a tree is equal to the height of the root

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knnn ,...,, 21 in1+in ki <≤1

Example: Unix Directory

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Binary Trees

v  A tree in which no node can have more than two children.

v  The depth of an “average” binary tree is considerably smaller

than N, even though in the worst case, the depth can be as large as N-1.

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An Example: Expression Trees

v Leaves are operands (constants or variables) v The other nodes (internal nodes) contain operators v Will not be a binary tree if some operators are not

binary Lecture 10 ECE250 9

Tree Traversal v Traversal is the process of visiting every node once

Ø  Visiting a node entails doing some processing at that node, but when describing a traversal strategy, we need not concern ourselves with what that processing is

v Three recursive techniques for tree traversal Ø  the left subtree is traversed recursively Ø  the right subtree is traversed recursively Ø  the root is visited

v What distinguishes the techniques from one another is the order of those 3 tasks

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Pre-order Traversal

v Node, Left, Right v Prefix Expression

Ø ++a*bc*+*defg

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The Pre-order Directory Listing

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Post-order Traversal

v Left, Right, Node v Postfix Expression

Ø abc*+de*f+g*+

Lecture 10 ECE250 13

In-order Traversal

v Left, Node, Right v Infix Expression

Ø a+b*c+d*e+f*g

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Algorithms

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Lecture 10 ECE250 16

Binary Tree ADT v BinTree ADT:

Ø  Accessor functions: §  key():int §  parent(): BinTree §  left(): BinTree §  right(): BinTree

Ø  Modification procedures: §  setKey(k:int) §  setParent(T:BinTree) §  setLeft(T:BinTree) §  setRight(T:BinTree)

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