1. COSC2100: Data Structures and Algorithms
Tree Basics and Binary Tree

2. Review So far, we have learnt two fundamental data storage structures
Array
Linked List
An ordered array
Search a key in the array: O(lgN) with binary search
Insert or delete a new key: O(N) worst case
Linked list
Search a key in the list: O(N) worst case
Insert a key: O(1)

3. Overview
We are going to learn another fundamental data storage structure
Binary Tree
With a special binary tree, Balanced Binary Search Tree
Search: O(lgN)
Insert and Delete: O(lgN)
The basics about trees
Terminology
Tree traversals

4. Tree Structure in Real Life
College
College of Art and Science
Departments
BSCI
MSCS
CHEM
Course types
MATH
COSC
COSC courses
COSC1000
COSC2100
COSC3100
Other examples:
Family tree
Table of contents
Company organization chart

5. A file system that stores documentations for several courses
Trees
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Trees in Computers
/usr/rt/courses
cosc1020
cosc2100
cosc3100
programs
homework
lectures
A file system that stores documentations for several courses
Trees

6. Tree Data Structure A tree consists of nodes connected by edges
Trees
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Tree Data Structure
A tree consists of nodes connected by edges
Nodes are represented as circles in the graph
Edges are lines in the graph
Trees are hierarchical structures B C A D E G H X Y Z
Trees

7. Tree Terminology I Root: the node at the top of the tree
Trees
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Tree Terminology I A B D C G H E F I J K
Root: the node at the top of the tree
Node A
Edge: directed connection from a node’s parent to the node
Edges from A to B, from B to F, from F to K
Parent: the node to which the given node is connected via the upward edge
B’s parent: A
A’s parent: null
Child: the node to which the given node is connected via downward edges
A’s child: B, C, D
C’s child: G, H
Trees

8. Tree Terminology II Internal node: node with at least one child
Trees
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Tree Terminology II
Internal node:
node with at least one child
Nodes A, B, C, F
leaf node
node without children
Nodes E, I, J, K, G, H, D
Siblings of a node
The nodes with the same parent
I’s siblings: J and K
Subtree (CGH):
tree consisting of a node and its children, children’s children…
Root of the subtree: C
Edge from A (C’s parent) to C
subtree A B D C G H E F I J K
Trees

9. Trees
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Tree Terminology III
Path: a path form node n1 to nk is defined as a sequence of nodes n1, n2, …, nk such that ni is the parent of ni+1 for 1 <= i < k
Length: the length of this path is the number of edges on the path
The depth of a node ni is the length of path form the root to ni.
The depth of the root is 0
The depth of the tree is the largest depth among leaves.
The height of ni is the length of the longest path from ni to a leaf.
All leaves are at height 0.
The height of a tree is the height of the root.
Trees

10. Examples Exercises: Parent of J: children of J:B D C G H E F I J K
Exercises:
Parent of J:
children of J:
Path from A to F and its length:
Height and depth of F:

11. More Exercises Which node is the root? Which nodes are leaves?
The tree’s depth?
For node E, list its
parent
children
siblings
height
depth B C D E F G H I J K L M

12. Yet More Exercises Which node is the root? Which nodes are leaves?
The tree’s depth?
For node A, list its
parent
children
siblings
height
depth A

13. Trees
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Binary Trees A
A binary tree is a tree in which every node has at most two children
left child and
right child
A node in a binary tree doesn’t necessarily have two children
only a left child
only a right child
or no children at all B C D E G H I
Trees

14. Representing Tree in Java Code
Trees
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Representing Tree in Java Code
class Node { int iData; Node left; Node right; //constructor //display }
class Tree {
private Node root;
//constructor
//methods for insertion, deletion… }
Node is the building blocks for Tree, just as Link for LinkList
Node root is the only entry to a Tree object, as Link first to a LinkList object
Trees

15. Traversing a Tree
To traverse a tree means to visit all the nodes in some specified order
Inorder traversal
Preorder traversal
Postorder traversal

16. Trees
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Inorder Traversal
Algorithm inOrder(v)
If v==null
Return
inOrder (left (v))
System.out.print(v.iData + “ “)
inOrder (right (v))
A node is visited after its left subtree and before its right subtree
What is printed for the given tree if executing
inOrder(root) 6 2 8 1 4 7 9 3 5
Trees

17. Preorder Traversal A node is visited before its children
Trees
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Preorder Traversal
A node is visited before its children
If visiting a node is printing the string, what will be printed for the given tree?
Algorithm preOrder(v)
visit(v)
for each child w of v
preorder (w) 1
Make Money Fast! 2 5
1. Motivations
2. Methods 7 6 3 4
2.1 Stock Fraud
2.3 Bank Robbery
1.1 Greed
1.2 Avidity
Trees

18. Postorder Traversal A node is visited after its descendants
Trees
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Postorder Traversal
A node is visited after its descendants
Application: compute space used by files in a directory and its subdirectories
Demonstrate how it works.
Algorithm postOrder(v)
for each child w of v
postOrder (w)
visit(v) 7
cs16/ 3 6
homeworks/
programs/ 1 2 4 5
h1c.doc 3K
h1nc.doc 2K
DDR.java 10K
Robot.java 20K
Trees

19. Pop Quiz A
By visiting the node, we print the character stored in the node
Show the result of
Preorder
Inorder
Postorder
of the given tree. B C D E F G H I J K L M

20. Summary Tree Terminologies Tree traversals Hierarchical data structure
Root
Internal, external (leaf) nodes
Parent
Depth
Height
Path
Tree traversals
Preorder, inorder, and postorder