2015-T2 Lecture 20 School of Engineering and Computer Science, Victoria University of Wellington  Marcus Frean, Lindsay Groves, Peter Andreae, John Lewis, and Thomas Kuehne, VUW COMP 103 Thomas Kuehne Introduction to Trees

2 RECAP  Used linked lists to implement linear data structures  Efficiency issues still remain TODAY  Introduction to Trees  Reading: Chapter 16 in textbook

3 Remaining Efficiency Challenge  Linear Linked Structures (LinkedList, LinkedStack, … )  adding / removal operations are O(1)   but random access is expensive   Why?  reducing a search or access problem by 1 and leaving a subproblem of size n-1 is not a good divide & conquer strategy  This is why a naïve QuickSort implementation can be slow (O(n 2 ) in the worst case)

4 Divide & Conquer  Challenge  Guess the secret animal with as few questions as possible  Strategy  eliminate as many as possible in each step Toby Tiger Lea Lion Bully Bulldog Carrie Collie Tanja Tui Kurt Kaka Tim Turtle Sally Snake Egg Laying Mammal Reptile Bird 

5 Divide & Conquer  Linear access is slow  only one candidate eliminated at a time Toby Tiger Lea Lion Bully Bulldog Carrie Collie Tanja Tui Kurt Kaka Tim Turtle Sally Snake

6 Divide & Conquer  Linear access is slow  only one candidate eliminated at a time  Hierarchical access is fast  many (proportional to total amount) eliminated at a time Toby Tiger Lea Lion Bully Bulldog Carrie Collie Tanja Tui Kurt Kaka Tim Turtle Sally Snake Egg Laying Mammal Reptile Feline Canine Bird

7 From Linear to Hierarchical Access  Linear linkage  structure split into one head and the rest Toby Tiger Lea Lion Bully Bulldog Carrie Collie Tanja Tui Kurt Kaka Tim Turtle Sally Snake

8 From Linear to Hierarchical Access  Hierarchical linkage  structure split into parts, each containing multiple elements Toby Tiger Lea Lion Bully Bulldog Carrie Collie Tanja Tui Kurt Kaka Tim Turtle Sally Snake Egg Laying Mammal Reptile Feline Canine Bird Animal

9 From Linear to Hierarchical Access  Hierarchical linkage  structure split into parts, each containing multiple elements Toby Tiger Lea Lion Bully Bulldog Carrie Collie Tanja Tui Kurt Kaka Tim Turtle Sally Snake Egg Laying Mammal Reptile Feline Canine Bird Animal

10 Trees are Hierarchical Structures  Upside Down Trees? Feline Leo Lion Toby Tiger Bully Bulldog Carrie Collie Tanja Tui Kurt Kaka Tim Turtle Sally Snake Egg Laying Mammal Reptile Canine Bird Animal

11  Some Terminology Trees are Hierarchical Structures Feline Leo Lion Toby Tiger Bully Bulldog Carrie Collie Tanja Tui Kurt Kaka Tim Turtle Sally Snake Egg Laying Mammal Reptile Canine Bird Animal root leaves branch

12 Trees are Hierarchical Structures  Same Terminology, despite different orientation Feline Leo Lion Toby Tiger Bully Bulldog Carrie Collie Tanja Tui Kurt Kaka Tim Turtle Sally Snake Egg Laying Mammal Reptile Canine Bird Animal root leaves branch  Implementation with LinkedNode++  support multiple successors, instead of just one

13 Feline Leo Lion Toby Tiger Bully Bulldog Carrie Collie Tanja Tui Kurt Kaka Tim Turtle Sally Snake Egg Laying Mammal Reptile Canine Bird Animal Implementation as a Linked Structure  Implementation with LinkedNode++  support multiple successors, instead of just one

14  Generalised LinkedNode Representing trees in Java M Linked List Nodes FC

15  Generalised LinkedNode Representing trees in Java Binary Tree Nodes M FCTL

16  Generalised LinkedNode Representing trees in Java General Tree Nodes F … … … C … … … M … … … T … … … L … … … some collection type (ordered or unordered)

17  Arrays  It is possible to represent trees with arrays  No reference overhead!  Clever assignment of nodes to array indeces  We’ll probably won’t cover this (outside heaps)  Textbook → Ch Representing trees in Java

18 More Tree Examples  Other Taxonomies  e.g. game genres  Organisational Charts  CEO, managers, employees, slaves, …  Filing systems  e.g., the folder structure of your hard drive  Computer Graphics models   Octrees, for partitioning 3D space hierarchical structures naturally represented with trees (rather than using trees as an access technique) hierarchical structures naturally represented with trees (rather than using trees as an access technique)

19  Other Taxonomies  e.g. game genres  Organisational Charts  CEO, managers, employees, slaves, …  Filing systems  e.g., the folder structure of your hard drive  Computer Graphics models   Octrees, for partitioning 3D space More Tree Examples

20 More Tree Examples  Other Taxonomies  e.g. game genres  Organisational Charts  CEO, managers, employees, slaves, …  Filing systems  e.g., the folder structure of your hard drive  Computer Graphics models   Octrees, for partitioning 3D space  Decision processes ……

21 Planning  Tic Tac Toe  search tree for moves XXX XXX … OXOXX O X O X O X O … often not represented explicitly; only implicitly “created” by recursion

22 More Tree Terminology  A tree is a collection of items in a strict hierarchical structure.  Each item is in a node of the tree.  The root is at the top.  The leaves are at the bottom.  Each node may have child nodes – except a leaf, which has none.  Each node has one parent - except the root, which has none.  An edge joins a node to its parent – may be labelled.  A subtree is a node plus all its descendents.  The depth of a node is its distance from the root.  The height or depth of a tree is the depth of the lowest leaf.  Level = set/list of nodes at the same depth.  Branch = sequence of nodes on a path from root to a leaf. Children may be ordered/unordered Tree may or may not store explicit parent references

23 Terminology visualised K K G G C C I I M M Q Q O O A A E E node, root, leaf, child, parent, edge, subtree, depth, height, level, branch, siblings, ancestors, descendants, cousins, …