Computer Science 112

Fundamentals of Programming IIIntroduction to Trees

What Is a Tree?

• Each item has at most one predecessor

• Each item can have many successors

What Is a Tree?

• Start with a single node, called the root

Rootnode

What Is a Tree?

• Start with a single node, called the root

• Each successor node is a child or daughter node

Rootnode

Daughters of root node

What Is a Tree?

• Successors are also called descendants

Rootnode

Descendants of root node

What Is a Tree?

• Nodes without successors are called leaf nodes

• The set of leaf nodes is the frontier

Rootnode

Leaf nodes (the frontier)

What Is a Tree?

• Nodes with at least one successor are called interior nodes

Rootnode

Interior nodes

What Is a Tree?

X

• Predecessors are also called ancestors

• The immediate predecessor is called the parent

Rootnode

Ancestors of node X

What Is a Tree?

• Nodes with the same parent are called siblings

Rootnode

Siblings

What Is a Tree?

• Levels in a tree are numbered from 0

Rootnode

Level 0

Level 1

Level 2

Level 3

There are three nodes in level 3

What Is a Tree?

• A binary tree allows at most two successors per node

Rootnode

What Is a Tree?

• A general tree allows any number of successors per node

Rootnode

Trees as Recursive Data Structures

• A tree is either– empty, or– consists of a node containing

• a datum

• one or more subtrees

Each subtree is itself another tree

Tree Traversals

• We’d like to visit each data item in a tree

• Are the items randomly ordered, as in a bag or set?

• Think of visiting the data in a node, and its left and right subtrees, in some order

Order of nodes visited:

D B

C

F

A E

D

G

Preorder Traversal

Visit the dataVisit the left subtreeVisit the right subtree

Order of nodes visited:

D B B

C

F

A E

D

G

Preorder Traversal

Visit the dataVisit the left subtreeVisit the right subtree

Order of nodes visited:

D B A B

C

F

A E

D

G

Preorder Traversal

Visit the dataVisit the left subtreeVisit the right subtree

Order of nodes visited:

D B A CB

C

F

A E

D

G

Preorder Traversal

Visit the dataVisit the left subtreeVisit the right subtree

Order of nodes visited:

D B A C FB

C

F

A E

D

G

Preorder Traversal

Visit the dataVisit the left subtreeVisit the right subtree

Order of nodes visited:

D B A C F EB

C

F

A E

D

G

Preorder Traversal

Visit the dataVisit the left subtreeVisit the right subtree

Order of nodes visited:

D B A C F E GB

C

F

A E

D

G

Preorder Traversal

Visit the dataVisit the left subtreeVisit the right subtree

Order of nodes visited:

AB

C

F

A E

D

G

Inorder Traversal

Visit the left subtreeVisit the dataVisit the right subtree

Order of nodes visited:

A BB

C

F

A E

D

G

Inorder Traversal

Visit the left subtreeVisit the dataVisit the right subtree

Order of nodes visited:

A B CB

C

F

A E

D

G

Inorder Traversal

Visit the left subtreeVisit the dataVisit the right subtree

Order of nodes visited:

A B C DB

C

F

A E

D

G

Inorder Traversal

Visit the left subtreeVisit the dataVisit the right subtree

Order of nodes visited:

A B C D EB

C

F

A E

D

G

Inorder Traversal

Visit the left subtreeVisit the dataVisit the right subtree

Order of nodes visited:

A B C D E FB

C

F

A E

D

G

Inorder Traversal

Visit the left subtreeVisit the dataVisit the right subtree

Order of nodes visited:

A B C D E F GB

C

F

A E

D

G

Inorder Traversal

Visit the left subtreeVisit the dataVisit the right subtree

Order of nodes visited:

AB

C

F

A E

D

G

Postorder Traversal

Visit the left subtreeVisit the right subtreeVisit the data

Order of nodes visited:

A CB

C

F

A E

D

G

Postorder Traversal

Visit the left subtreeVisit the right subtreeVisit the data

Order of nodes visited:

A C BB

C

F

A E

D

G

Postorder Traversal

Visit the left subtreeVisit the right subtreeVisit the data

Order of nodes visited:

A C B EB

C

F

A E

D

G

Postorder Traversal

Visit the left subtreeVisit the right subtreeVisit the data

Order of nodes visited:

A C B E GB

C

F

A E

D

G

Postorder Traversal

Visit the left subtreeVisit the right subtreeVisit the data

Order of nodes visited:

A C B E G FB

C

F

A E

D

G

Postorder Traversal

Visit the left subtreeVisit the right subtreeVisit the data

Order of nodes visited:

A C B E G F DB

C

F

A E

D

G

Postorder Traversal

Visit the left subtreeVisit the right subtreeVisit the data

Order of nodes visited:

D B F A C E GB

C

F

A E

D

G

Level Order Traversal

For each level Visit data from left to right

Tree Applications

• File directories

• Processing sentences (computer programs or natural languages)

• Searchable data structures

• Heaps (implement heap sort, priority queues)

For Wednesday

Binary Search Trees