1. COMP 103
Introduction to Trees

2. RECAP-TODAY RECAP TODAY Linked List, Linked Stack, Linked Queue
Other linked collections
TODAY
Introduction to Trees
Reading: Chapter 16 in textbook

3. Other linked collections
LinkedStack and LinkedQueue
All the key operations are O(1)
Set
Can store sorted or unsorted
What is the cost of add, remove, contains, size, equals in each case?
Bag
Same as set, but add a count to each node.
Map
Same as set, but store value as well as key in each node
Or store pointed to a key-value pair
What is the cost of get, put, remove?

4. Linear vs. Hierarchical Structures
So far we’ve looked at unordered structures and linear structures:
Unordered structures have no structural order between items:
Set, Bag, Map
Linear structures have items arranged in order one after another
List, Stack, Queue…
Some data have a hierarchical or ‘tree-like’ structure
Hierarchical structure have items arranged in levels, with items ‘above’ or ‘below’ the other, such as
Family tree
Organisational Charts
Taxonomies
Filing systems (paper-based, computer-based)
Models of shapes for computer graphics
Programs/decision processes…. And so on….

5. Trees
We're going to look into the data structure and the crucial algorithms for trees
Applications of tree structures:
Representing data with a natural hierarchical structure
Using trees as data structures to implement other collections: sets, maps and priority queues (and another sorting algorithm!)
Note: some data has an even more complex network-like structure:
Communications networks
Geographical maps (real world, or in computer games)

6. Family Trees: ancestral tree
Justine
Jeremy
Julie
John
Julia
Jesse
Jordan
Jenna
Jack
Jacob
Jules
James
Jada
Jenny
Jackie
Jasmine
Jean
seniority

7. Organisational hierarchy
Jane
Jeremy
Julie
Justin
John
Jules
Jenny
Jesse
Jordan
Jacky
Jenna
Juan
Jacob
Julia
James
Jada
Jared
Jake
Joseph
Jasmine
seniority

8. Taxonomy Animals Reptiles Mammals Primates Felines Canines Turtles
Snakes
Lizards
Tiger
Leopard
Lion
Cat

9. Program Structure
private void reportMachineQueues(int tick) { System.out.printf("%4d: ", tick); for (Queue<Job> queue : machineQueues) { if (queue.isEmpty()) System.out.print(" - "); else System.out.printf("%3s ", queue.peek().getID()); } System.out.println(); if (verbose) for (int m=0; m<NumberOfMachines; m++) { Queue<Job> queue = machineQueues.get(m); System.out.printf(" M%d: %d jobs: ", m, queue.size()); boolean first = true; for (Job job : queue) { if (first) { System.out.print(job); first=false; else System.out.printf("\n %s", job);

10. Tracing program execution
archWall(10, 300, 80, 40)
aw(50,220,40,20)
drawArch
aw(10, 220, 40, 20)
drawArch
aw(10,180,20,10)
aw(30,180,20,10)
drawArch
aw(50,180,20,10)
aw(70,180,20,10)
drawArch
drawArch
drawArch
drawArch

11. Noughts and Crosses (Tic Tac Toe)X X X X X O X O X X X X O O O

12. Decision Tree

13. 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.
This is more correctly called a rooted, directed, labelled tree.
Children may be ordered/unordered
Unique!!

14. Components of Trees A C B E F G J K M
Node, root, leaf, child, parent, edge, subtree, depth, height, level, branch,
siblings, ancestors, descendants, cousins, …

15. Common Constraints on Trees
Number of children per node (“arity”)
2: Binary tree
3: Ternary tree
N: N-ary tree (5-ary, 100-ary, …)
Strict N-ary: every non-leaf has exactly N children; otherwise at most N
General Tree: No limit on number of children
Ordering constraints:
Children of a node ordered (List of children).
Children of a node unordered (Set/Bag of children).
Item in each node is less than items in child nodes.
Item at each node is smaller that all items in its left subtree and greater than all items in its right subtree.
Balancing:
Subtrees are the same height (all leaves are on same level)
Subtrees heights differ by at most one (leaves within one level of each other)
What is a unary tree?

16. What can you do with a tree?
Construct/modify/print it:
Add nodes, remove nodes (where?)
Parsing: read a string and turn it into a tree, eg. program, expression
Search it:
Find a node with a given value/property
Find a path to (or from) a node
Compute some property of the tree or its data:
Find the height, width, arity, …
Sum, max, average, … of values stored
Find a given person’s mother, or all of their grandchildren
Find the manager with the most (direct/indirect) underlings

17. Traversing trees
Many tree algorithms involve doing something at each node in a tree
Different algorithms require nodes to be visited in different orders
Breadth-first/level-order:
Visit nodes in order of increasing depth
Root, then all nodes with depth 1, all with depth 2, …
May be in specified order within each level
Depth-first:
Visit one child and all of its descendants before its siblings
May visit root before, after or between its children
May visit children in specified order

18. Trees and Recursion Recursively defined data structure:
A tree is a node, along with a set/list of subtrees (empty for leaf)
Can a tree be empty?
Recursion is very natural for trees – important!
If you don’t use recursion you often need to simulate it (later)

19. Binary Trees
Each node may have a left child and/or a right child
Notice that tree size is limited by the depth of the tree. For binary tree, nodes at level k = n, for N-ary tree, k = n ^ (k-1). Size for binary is: … + n = 2^n – 1 and size for N-ary is (n^k-1)/(n-1) where k is number of full levels.
What’s the maximum number of nodes in a binary tree of height k ?
What’s the minimim height of a binary tree with n nodes?

20. Representing tree data in Java
What kind of data structure can we use to represent a tree?
Linked Structures, just like Linked List Nodes:
Binary Tree Nodes: General Tree Nodes
Arrays ?!
we'll come to this later on... G T K M Z A G M K G K M