1. Data Structures and Design in Java © Rick Mercer
Chapter 18
Binary Trees
Data Structures and Design in Java © Rick Mercer

2. A 3rd way to structure data
We have considered two data structures
Arrays
Singly linked
Both are linear
each node has one successor and one predecessor (except the first and last)
We now consider a hierarchical data structure where each node may have two successors, each of which which may have two successors, and so on

3. Trees in General
A tree has a set of nodes and directed edges that connect them
One node is distinguished as the root
Every node (except the root) is connected by exactly one edge from exactly one other node
A unique path traverses from the root to each node
Root

4. Some tree terminology
Node An element in the tree references to data and other nodes
Path The nodes visited as you travel from root down
Root The node at the top It is upside down!
Parent The node directly above another node (except root)
Child The node(s) below a given node
Size The number of descendants plus one for the node itself
Leaves Nodes with no children
Height The length of a path (number of edges, -1 for empty trees)
Levels The top level is 0, increases
Degree The maximum number of children from one node

5. Trees used in many ways Hierarchical files systems
In Windows, \ represents an edge (or / in Linux)
Each directory may be empty, have children, some of which may be other directories
A tiny bit of MAC's file system

6. A few uses of trees Implement data base systems
Store XML data from a web service as a Java collection DOM tree
Binary Search Trees insert, remove, and find are O(log n) 56
/ \
25 80
/ \ \
Compilers: Expression tree, Symbol tree
Store game of 20?s
Once used as our logo

7. The Binary Tree
Structure has nodes to store an element along with the left and right children (binary trees)
root is like first, front, or top in a singly linked structure
nodes
root
"T"
"L"
"R"
edges

8. Binary Trees
A binary tree is a tree where all nodes have zero, one or two children
Each node is a leaf (no children), has a right child, has a left child, or has both a left and right child A B C D F D

9. Application: Huffman Tree
Binary trees were used in a famous first file compression algorithm Huffman Tree
Each character is stored in a leaf
Follow the paths
0 go left, 1 go right
a is 01, e is 11
What is t?
What is
31 bits vs. 12*8 = 96 bits
'a'
' '
'e'
't'
'h'
'r'

10. Application: Binary Search Trees
Insert, Search, Remove Operations:O(log n) 50 25 75 12 35 66 90 28 41 54 81 95

11. Binary Search Trees
A Binary Search Tree (BST) data structure is a binary tree with an ordering property
BSTs are used to maintain order and faster retrieval, insertion, and removal of individual elements
A Binary Search Tree (BST) is
an empty tree
consists of a node called the root, and two children, left and right, each of which are themselves binary search trees. Each BST contains a key at the root that is greater than all keys in the left BST while also being less than all keys in the right BST. Key fields are unique, duplicates not allowed.

12. Are these BSTs? only the keys are shown50
Is this a BST? 25 75 12 45 66 90 50
Is this a BST? 25 75 12 55 73 90

13. BST Algorithms only the keys are shown
Think about an algorithm that retrieves a node
It is similar to binary search.
Start at root, go left or right until?_____
(Ex: target == 65)
Assuming the BST is "balanced", what is the tightest upper bound in Big-O? 50 25 75 12 36 65 90

14. values must be unique
insert returns false if the key exists
In this case, the mapping is not added to the collection

15. insert only the keys are shown
Start at the root and go either left or right to find the insertion point.
root
tracker 50 25 75 12 36 65 90
To insert 28: 28 < 50 so go left; 28 > 25 so go right;
left link from 36 is null so stop

16. Adding the new node
Replace the left child of 36, which was null, with a new BST node with data == 28 50 25 75 12 36 65 90 28