cs-2852 data structures lecture 11 andrew j. wozniewicz image copyright © 2010 andyjphoto.com

CS-2852Data StructuresLECTURE 11

Andrew J. Wozniewicz

Image copyright © 2010 andyjphoto.com

CS-2852 Data Structures, Andrew J. Wozniewicz

Agenda• Tree Overview • Binary Tree– Binary Search Tree

• Searching• Traversals– Pre-Order– In-Order– Post-Order


Trees

Leaves

Branches

Root

COMPUTER SCIENCE

Root

Branches

Leaves

NATURE


Trees

COMPUTER SCIENCE

Leaves

Branches

Root

What is a Tree?Vice-

President

Team Lead

SE1

SE2

SE3

Team Lead

SE4

SE5

Manager

BA1

BA2

Consultant


President

Team Lead

SE1

SE2

SE3

Team Lead

SE4

SE5

Manager

BA1

Consultant Secretary

ROOT


President

Team Lead

SE1

SE2

SE3

Team Lead

SE4

SE5

Manager

BA1


ROOT

CHILD


President

Team Lead

SE1

SE2

SE3

Team Lead

SE4

SE5

Manager

BA1


ROOT

CHILD

LEAVES


What is a Tree?

• There is exactly one (i.e. unique) path from the root to each node.

• There is exactly one path from each node to the root.

• There is exactly one path between any nodes.• Each node can have arbitrarily many children.

Each Child has exactly one Parent


What is a Tree?• Like Linked Lists, Trees are “linked”

data structures.• Hierarchical, rather than linear

chaining.• A node can be a “child” of another

node; the only node that is not a “child” is the “root”.

• Nodes with children are referred to as “Parents”.


Height of a Tree

• Depth (level): Measures the distance from its root– Root node has level 1– Any other node’s level is the level of its

parent + 1

Number of nodes in the longest path from the root node to a leaf node.

DEFINITION


Tree Terminology• Node (External/Internal)• Root, Parent, Child• Ancestor, Descendant, Sibling• Branch, Subtree

Binary Tree

• Hierarchy of data with some constraints

• A Root node• 0-2 children– Left Child– Right Child

• Each child is itself a tree


(Binary) Tree Node

protected static class Node<E> { protected E data; protected Node<E> left; protected Node<E> right; public Node(E data) { this.data = data; this.left = null; this.right = null; } }

What does it resemble?


Binary Search Tree• Same structural

rules as Binary Tree…

• A sorted hierarchy of data – Left child less than

parent– Right child greater

than parent

1 3

2

4

5 7

6

261375


Adding Data• Recursive algorithm:• Case 1: Empty Tree

– Becomes the root node• Case 2: Smaller Value

– Recursively add to left• Case 3: Larger Value

– Recursively add to right• Case: Equal Value

– Treat as larger value 1

2

4

7

6

2617

4

4


Searching

Node find(Node root, E value) { if (root == null)

return null; if (root.value==value) return root; if (value < root.value) return find(current.left, value); return find(current.right,value);}

1 3

2

4

5 7

6

theRoot

• find(theRoot, 3);• find(theRoot, 5);• find(theRoot, 8);


Advantages of Trees• Don’t ever have to examine every

node to determine if a value is in.– Unlike in a Linked List

• Only looking at a subset of the data in the tree

• The largest possible number of comparisons is equal to the height of the tree.


Full, Perfect, and Complete BT

• Full:– All nodes have either 0 or 2 children

• Perfect– Full, with 2height-1 nodes

• Complete– Perfect through level height-1


Linked Structure Traversals

• Linked List, Array: – Left-to-Right– Right-to-Left

• Stack– Last-In, First-Out

• Queue– First-In, First-Out

• Tree?– Left-First?– Right-First?

1 3

2

4

5 7

6

theRoot


Tree Traversals• Enumerate nodes in a well-known

order…• Basic algorithm:

–Process Current Node–Visit Left–Visit Right

• What varies is the order No

parti

cula

r Ord

er Im

plie

d


Three Common Traversal Orders

• Pre-Order– Process Current– Visit Left Child– Visit Right Child

• In-Order– Vist Left Child– Process Current– Visit Right Child

• Post-Order– Visit Left Child– Visit Right Child– Process Current Child


Pre-Order Traversalvoid visit(Node current) { if (current==null) return; process(current.value); visit(current.left); visit(current.right);}

2 8

1 3 6

5 7

4

4 2 1 3 8 6 5 7


In-Order Traversalvoid visit(Node current) { if (current==null) return; visit(current.left); process(current.value); visit(current.right);}

2 8

1 3 6

5 7

4

1 2 3 4 5 6 7 8


Post-Order Traversalvoid visit(Node current) { if (current==null) return; visit(current.left); visit(current.right); process(current.value);}

2 8

1 3 6

5 7

4

1 3 2 5 7 6 8 4


Summary• What is a Tree? • Binary Tree– Binary Search Tree

• Searching• Traversals– Pre-Order– In-Order– Post-Order

cs-2852 data structures lecture 11 andrew j. wozniewicz image copyright © 2010 andyjphoto.com

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