ECE250: Algorithms and Data Structures Trees

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1 ECE250: Algorithms and Data Structures Trees Ladan Tahvildari, PEng, SMIEEE Professor Software Technologies Applied Research (STAR) Group Dept. of Elect. & Comp. Eng. University of Waterloo Materials from CLRS: Chapter 10.4 and Appendix B.5 Weiss: Chapter 4

2 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 Lecture 10 ECE250 2

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)? Lecture 10 ECE250 3

4 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 T 1, T 2,..., T k, each of whose roots are connected by a directed edge from r Lecture 10 ECE250 4

5 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

6 Some Terminologies v Path Ø A sequence of nodes 1 such that i is the parent of n for v Length i+1 1 i < k 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 n n,...,, 2 Ø The height of a tree is equal to the height of the root Lecture 10 ECE250 6 n k n

7 Example: Unix Directory Lecture 10 ECE250 7

8 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. Lecture 10 ECE250 8

9 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

10 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 Lecture 10 ECE250 10

11 Pre-order Traversal v Node, Left, Right v Prefix Expression Ø ++a*bc*+*defg Lecture 10 ECE250 11

12 The Pre-order Directory Listing Lecture 10 ECE250 12

13 Post-order Traversal v Left, Right, Node v Postfix Expression Ø abc*+de*f+g*+ Lecture 10 ECE250 13

14 In-order Traversal v Left, Node, Right v Infix Expression Ø a+b*c+d*e+f*g Lecture 10 ECE250 14

15 Algorithms Lecture 10 ECE250 15

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) Root Lecture 10 ECE250 16

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