Binary Trees Review From CSCI-1321 Data Abstractions CSCI-2320 - PowerPoint PPT Presentation

Binary Trees Review From CSCI-1321 Data Abstractions CSCI-2320 Dr. Tom Hicks Computer Science Department

Root Root Node - A Node Without A Parent. Root Root 3

Leaf Leaf Node - A Node Without Children. Leaf Leaf 5

Ancestors {D,11} - Nodes In The Path(s) To The Root. B, F  Ancestors Of D 5, 2, 1  Ancestors Of 11 7

Descendants {B,6} - Nodes In The Path(s) To The Leaves. A,D,C,E  Descendants Of B 12,13,24,25,26,27  Descendants Of 6 9

Skew Trees Skew Tree - A Tree With No More Than 1 Node At Each Level. Empty Tree A Skew Tree Has The Same Performance As A Linked List! 11

Full/Perfect Binary Trees Full Tree - A Tree In Which The Lowest Level Is Fully Populated  Every Node, Other Than The Leaves, Has Two Children. Empty Tree Best Case Tree! Also Called A "Perfect" Tree 13

Height Of Tree Height Of A Tree Is The Number Of Edges In The Longest Path From The Root To A Leaf Node Height Height = 5 15

No Levels On A Tree Some Start With 0  Others With 1  WE ARE GOING TO USE 1 Level       1 Level      2 Height Level    3 Level   4 Level   5 Level          6 Height = 5 17

Perfect/Full Trees  NL  No Levels - 1 Max Nodes  Perfect Tree  NL = 2 2 NL - 1 = 2 2 - 1 = 3 2 NL - 1 = 2 3 - 1 = 7 19

Perfect/Full Trees  NL  No Levels - 2 Max Nodes  Perfect Tree  NL = 4 2 NL - 1 = 2 4 - 1 = 15 Height Of Perfect Binary Tree With 15 Nodes = Log 2 (15) = ~4 20

Perfect/Full Trees  NL  No Levels - 3 Max Nodes  Perfect Tree  NL = 5 2 NL - 1 = 2 5 - 1 = 31 Height Of Perfect Binary Tree With 15 Nodes = Log 2 (31) = ~5 21

Average Search = Total Searches / No Nodes Compute # Searches         1 1      4 2 2     3 3 3 9      12 4 4 4 ___ 26 26/9 = 2.89 23

Balanced Binary Tree A Balanced Tree is one that cannot be expressed in fewer levels; it will have the lowest Average Search for that number of Nodes. 25

Inorder Traversal Left  Root  Right  There Are Recursive & Non-Recursive Solutions! Preorder Traversal Root  Left  Right  There Are Recursive & Non-Recursive Solutions! Postorder Traversal Left  Right  Root  There Are Recursive & Non-Recursive Solutions! 26

Recursive Solutions Are Quite Easy To Code Traverse The Left Visit The Root Traverse The Right 27

Inorder Traversal D  B  H  E  I  A  F  C  G 28

Inorder Traversal D  B  H  E  I  A  F  C  G H  D  I  B  J  E  K  A  L  F  C  G 29

Preorder Traversal A  B  D  E  H  I  C  F  G A  B  D  H  I  E  J  K  C  F  L  G 30

Postorder Traversal D  H  I  E  B  F  G  C  A H  I  D  J  K  E  B  L  F  G  C  A 31

Data Abstractions CSCI 2320 Dr. Thomas E. Hicks Computer Science Department Trinity University Textbook: Introduction To Algorithms 3 rd Edition By Cormen, Leiserson, Rivest, Stein Textbook: A Tour Of C++ By Bjarne Stroustrup Special Thanks To For MIT Press & Adison Wesley For Content & Graphics That Are Relevant To This Presentation. 32

Average Search = Total Searches / No Nodes 100 150 50 125 175 75 25 40 25 40 50 75 100 125 150 175