Trees Carlos Delgado Kloos M Carmen Fernndez Panadero Raquel M. - - PowerPoint PPT Presentation

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Trees Carlos Delgado Kloos M Carmen Fernndez Panadero Raquel M. - - PowerPoint PPT Presentation

Nov 06, 2022 258 likes •952 views

Trees Carlos Delgado Kloos M Carmen Fernndez Panadero Raquel M. Crespo Garca Dep. Ingeniera Telemtica Univ. Carlos III de Madrid cdk@it.uc3m.es cdk@it.uc3m.es Java: Trees / 1 Contents Concept Implementation Non

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cdk@it.uc3m.es cdk@it.uc3m.es Java: Trees / 1

Trees

Carlos Delgado Kloos Mª Carmen Fernández Panadero Raquel M. Crespo García

  • Dep. Ingeniería Telemática
  • Univ. Carlos III de Madrid
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Contents

 Concept

Non recursive definition Recursive definition Examples

 Terminology

Key concepts Properties

 Implementation

Sequence-based Linked structure

Basic operations Traversals

 Particular cases

Binary search trees Binary heaps

Java: Trees / 2 rcrespo@it.uc3m.es

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cdk@it.uc3m.es cdk@it.uc3m.es Java: Trees / 3

Quote

"The structure of concepts is formally called a hierarchy and since ancient times has been a basic structure for all western knowledge. Kingdoms, empires, churches, armies have all been structured into hierarchies. Tables of contents of reference material are so structured, mechanical assemblies, computer software, all scientific and technical knowledge is so structured..."

  • - Robert M. Pirsig: Zen and

the Art of Motorcycle Maintenance

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mcfp@it.uc3m.es Java: Trees / 4

Tress

Concept and characteristics

A tree is a non-linear data structure that stores the elements hyerarchically Trees can be defined in two ways:

Non-recursive definition Recursive definition

a e i j k b f g T1 T2

Recursive definition

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cdk@it.uc3m.es cdk@it.uc3m.es Java: Trees / 5

Non-recursive definition

A tree consists of a set of nodes and a set of edges, such that:

  • There is a special node called root
  • For each node c, except for the root, there is
  • ne edge to another node p (p is parent of

c, c is one of the children of p)

  • For each node there is a unique path

(sequence of edges) from the root

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cdk@it.uc3m.es Java: Trees / 6

Example

p c

Root (no parent) Hojas (sin hijos) Hojas (sin hijos) Leaves (no children) Hojas (sin hijos) Hojas (sin hijos) Leaves (no children) The parent of c A child of p sibling

  • f c
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cdk@it.uc3m.es Java: Trees / 7

Example

p c

Root (no parent) Hojas (sin hijos) Hojas (sin hijos) Leaves (no children) Hojas (sin hijos) Hojas (sin hijos) Leaves (no children) The parent of c A child of p sibling

  • f c
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mcfp@it.uc3m.es Java: Trees / 8

a b c d e f g h i j k

Trees

Recursive definition

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mcfp@it.uc3m.es Java: Trees / 9

a b c d e f g h i j k

Trees

Recursive definition

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mcfp@it.uc3m.es Java: Trees / 10

a b c d f g h i j k e

Trees

Recursive definition

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mcfp@it.uc3m.es Java: Trees / 11

a b c d f g h i j k e

Trees

Recursive definition

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mcfp@it.uc3m.es Java: Trees / 12

a b c d f g h i j k e

Trees

Recursive definition

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cdk@it.uc3m.es Java: Trees / 13

Recursive definition (1)

A tree is

  • A node
  • or a node and subtrees connected to the

node by means of an edge to its root

Doesn't include the empty tree

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cdk@it.uc3m.es Java: Trees / 14

Recursive definition (2)

A tree is

  • empty
  • or a node and zero or more non-empty

subtrees connected to the node by means of an edge to its root

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Examples

File system Structure of a book or a document Decision tree Arithmetic expressions

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Example

Expressions ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( ( ) ( ) ) ( ) )

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Terminology

A node is external, if it doesn't have children (it is a leaf) A node is internal, if it has one or more children A node is ancestor of another one, if it is its parent or an ancestor of its parent A node is descendent of another one, if the latter is ancestor of the former The descendents of a node determine a subtree where this node acts as the root

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Terminology

A path from one node to another one, is a sequence of consecutive edges between the nodes.

Its length is the number of edges it is composed of.

The depth of a node is the length of the path from the root to this node. The height of a tree is the depth of the deepest node. The size of a tree is the number of nodes.

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mcfp@it.uc3m.es Java: Trees / 19

a b c d e f g h i j k

Size of the tree: 11 Height of the tree: 3

Node Height Depth Size Internal / External

a

3 11

Internal

b

1 1 3

Internal

c

1 1

External

d

1 1 2

Internal

e

2 1 4

Internal

f

2 1

External

g

2 1

External

h

2 1

External

i

2 1

External

j

1 2 2

Internal

k

3 1

External

Example

Terminology and properties

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Terminology

Ordered tree

A tree is ordered, if for each node there exists a linear ordering for its children.

a b c

rcrespo@it.uc3m.es Java: Trees / 20

a c b

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A binary tree is an ordered tree, where each node has 0, 1 or 2 children (the left and the right child).

Terminology

Binary tree

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Basic algorithms

Size (number of nodes) Depth of a node Height Traversals

  • Euler
  • Pre-, in- and post-order

To simplify, we assume binary trees

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Implementations

Based on a linked structure Sequence-based

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Implementation based

  • n a linked structure

1 3 2 7 6 5 4

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Sequence-based implementation

1 3 2 7 6 5 4 1 2 3 4 5 6 7

p(rot)=1 p(x.left)=2*p(x) p(x.right)=2*p(x)+1

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public class BTree { protected BNode root; BTree() { root = null; } BTree(Object info){ root = new Bnode(info); }

Class Binary tree...

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…Class Binary tree...

public int size() { int size = 0; if (root != null) size=root.size(); return size; } public int height(){ int h = -1; if (root != null) h=root.height(); return h; }

cdk@it.uc3m.es Java: Trees / 27

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public void preorder() { if (root!=null) root.preorder(); } public void inorder() { if (root!=null) root.inorder(); } public void postorder() { if (root!=null) root.postorder(); } }

...Class Binary tree

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Class Binary node...

class BNode { private Object info; private BNode left; private BNode right; BNode() { this(null); } BNode(Object info) { this(info,null,null); } BNode(Object info, BNode l, BNode r) { this.info=info; left=l; right=r; }

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int size (){...;} int height (){...;} void preorder (){...;} void inorder (){...;} void postorder (){...;} }

...Class Binary node…

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BNode Class: Size

int size (){ int size = 1; if (left != null) size += left.size(); if (right != null) size += right.size(); return size; }

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BNode Class: Height

int height() { int hl = -1; int hr = -1; if (left !=null) hl = left.height(); if (right !=null) hr = right.height(); return 1 + Math.max(hl, hr); }

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Euler traversal

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Preorder traversal

First the node Then its children (recursively)

cdk@it.uc3m.es Java: Trees / 34

1 3 2 6 4 7 5

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Postorder traversal

7 6 1 5 3 4 2 First the children trees (recursively) Then the node

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Inorder (symmetric) traversal

2 5 1 7 3 6 4 First the left tree (recursively) Then the node Finally, the right tree (recursively)

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(A+B)*(C–D)

* – + D C B A

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Example

Infix Prefix Postfix A+B +AB AB+ A+B–C –+ABC AB+C– (A+B)*(C–D) *+AB–CD AB+CD–*

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Activity

Visualize expressions as trees http://www.cs.jhu.edu/~goodrich /dsa/05trees/Demo1/

cdk@it.uc3m.es Java: Trees / 39

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Postfix notation

HP calculators, RPN=Reverse Polish Not. Stack to store objects Eg: 3 5 + 6 2 – * 3 5 8 6 2 4 32

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Class BNode: preorder

void preorder (){ System.out.println(info); if (left != null) left.preorder(); if (right != null) right.preorder(); }

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Class BNode: postorder

void postorder (){ if (left != null) left.postorder(); if (right != null) right.postorder(); System.out.println(info); }

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Class BNode: inorder

void inorder (){ if (left != null) left.inorder(); System.out.println(info); if (right != null) right.inorder(); }

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Properties of binary trees

Let

  • E=Number of external nodes
  • I=Number of internal nodes
  • N=Size=E+I
  • H=Height

then

  • E=I+1
  • H+1≤E≤2H

H≤I≤2H-1 2*H+1≤N≤2H+1-1

  • log2(N+1)-1≤H≤(N-1)/2
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Binary search trees

A binary search tree is a binary tree where for each node n,

  • all the keys in the left subtree are

smaller (or equal) than the key of n

  • and all those of the right subtree larger

(or equal)

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Example

4 8 2 9 6 3 1 7 5

1 2 3

3

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Example

4 8 2 9 6 3 1 7 5

4 3 2 1

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Operations

Search Insertion Extraction

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Search

Searching “3”:

  • 3<4: go to left subtree
  • 3>2: go to right subtree
  • 3=3: element found

4 8 2 9 6 3 1 7 5 4 2 3

http://www.cosc.canterbury.ac.nz/ mukundan/dsal/BST.html

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Insertion

Inserting “6”:

  • 6<7: go to left subtree
  • 6>2: go to right subtree
  • when hole: insert

7 9 2 5 1 3 6 7 2 5

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Extraction (1)

Extracting “5”:

  • if leaf: extract
  • if degenerate: replace by child

7 9 2 5 1 3 3

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Extraction (2)

Extracting “2”:

  • if 2 children: replace by
  • largest at left, or
  • smallest at right subtree

7 9 2 5 1 3 3

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Activity

See animation of binary search trees

http://www.ibr.cs. tu-bs.de/courses/ ss98/audii/applets/ BST/BST-Example.html

cdk@it.uc3m.es Java: Trees / 53

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Heaps

A heap is a binary tree, where for each node n (except for the root), its key is larger or equal than the one of its parent. Utitity

Priority queues. Sorting algorithm

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Example

1 4 2 9 6 5 3 7 8

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Complete heap

1 4 2 9 6 5 3 7 8

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Sequence-based implementation

1 3 2 7 6 5 4 1 2 3 4 5 6 7

p(root)=1 p(x.left)=2*p(x) p(x.right)=2*p(x)+1

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Insert

4 6 5 20 7 9 15 25 16 12 14 8 11

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Insert

4 6 5 20 7 9 15 25 16 12 14 8 11 2

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Insert

4 6 5 2 7 9 15 25 16 12 14 8 11 20

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Insert

4 2 5 6 7 9 15 25 16 12 14 8 11 20

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Insert

2 4 5 6 7 9 15 25 16 12 14 8 11 20

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Extract

4 6 5 20 7 9 15 25 16 12 14 8 11

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Extract

8 6 5 20 7 9 15 25 16 12 14 11

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Extract

5 6 8 20 7 9 15 25 16 12 14 11

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Activity

Try out the form in

http://www.csse.monash.edu.au/~lloyd /tildeAlgDS/Priority-Q/

cdk@it.uc3m.es Java: Trees / 66

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Activity

Try out the applet in

http://www.cosc.canterbury.ac.nz/ mukundan/dsal/MinHeapAppl.html

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