Introduction to Trees

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Introduction to Trees

Data Structures

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Basic Tree Concepts

A tree consists of a finite set of elements, called nodes, and a finite set of directed lines, called branches, that connect the nodes.

Tree

 Nodes

 Branches

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Basic Tree Concepts (cons.)

Degree:

 Indegree When the branch is directed toward the node, it is indegree branch.

 Outdegree When the branch is directed away from the node, it is an outdegree branch.

Degree of the node: indegree + outdegree

AA

BB EE FF

CC DD GG HH II

The node B indegree = 1 The node B outdegree = 2 The node B degree =1+ 2 = 3

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Basic Tree Concepts (Cont.)

 Root

 If the tree is not empty, the first node is called the root.

 Indegree = ZERO

 With the exception of the root, all of the nodes in a tree must have an indegree of exactly one; that is, they may have only one predecessor.

AA

BB EE FF

CC DD GG HH II

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Basic Tree Concepts (Cont.)

 All nodes in the tree can have zero, one, or more branches

leaving them; that is, they may have outdegree of zero, one, or more.

 Leaf

 It is any node with an outdegree of zero, that is, a node with no successors.

AA

BB EE FF

CC DD GG HH II

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Basic Tree Concepts (Cont.)

 Internal node

 A node that is not a root or a leaf

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BB EE FF

CC DD GG HH II

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Basic Tree Concepts (Cont.)

Parent node

 It has a successor node, its outdegree is > 0

Child node? (vs.)

 It has a predecessor node, its indegree is 1

Siblings

 Two or more nodes with the same parent

AA

BB EE FF

CC DD GG HH II

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Basic Tree Concepts (Cont.)

 Ancestor

 Any node in the path from the root to the node

 Descendent

 Any node in the path below the parent node

 All nodes in the paths from a given node to a leaf node

AA

BB EE FF

CC DD GG HH II

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Basic Tree Concepts (Cont.)

Branch AF Branch

AF

Branch FI Branch FI

Parents: A, B, F

Children: B, E, F, C, D, G, H, I

Siblings: {B, E, F}, {C, D}, {G, H, I}

Leaves: C, D, E, G, H, I Internal nodes: B, F

AA

BB EE FF

CC DD GG HH II

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Basic Tree Concepts (Cont.)

Path

 Sequence of nodes in which each node is adjacent to the next node

Level

 Level of a node is its distance from the root

 The root has a zero distance from itself -> it is at level (0)

 Siblings levels are the same

AA

BB EE FF

CC DD GG HH II

Level 0 Level 1

Level 2

Path from the root to node I is AFI

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Basic Tree Concepts (Cont.)

Height

 Height of a tree=the level of the leaf of the longest path + 1

 The height of an empty tree is -1

 Depth == Height

AA

BB EE FF

CC DD GG HH II

Level 0 Level 1

Level 2

Path from the root to node I is AFI

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Basic Tree Concepts (Cont.)

Subtree

 Any connected structure below the root

 First node = root -> used to name the subtree

 Subtrees can be divided into subtrees

 A single node is a subtree

AA

BB EE FF

CC DD GG HH II

Subtree B

Subtree B Root of Subtree IRoot of Subtree I

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Recursive Definition of a tree

A tree is a set of nodes that either:

 Is empty

 Has a designated node, called the root, from which hierarchically

 descend zero or more subtrees, which are all subtrees

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Basic Tree Concepts (Cont.)

Tree Representation

 Tree generally implemented in the computer using pointers

 Outside the computer there are 3 representations

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Tree Representation:

1- General tree

Computer Computer

CaseCase

CPUCPU CD ROMCD ROM

ALUALU ROMROM

Floppy disk Floppy

disk

Controlle r Controlle

r

. . .

. . . . . .

. . .

• Chart format: the notation of figures

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Tree representation:

2- Indented list

Part Number Description

301 Computer

301-1 Case

. . . . . .

301-2 CPU

301-2-1 Controller

301-2-1 ALU . . . . . . 301-2-9 ROM

301-3 F Disk

. . . . . .

301-9 CD-ROM

. . . . . .

• Bill-of-materials systems, parts list represent the assembly structure of an item

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Tree representation 3- parenthetical listing

• Algebraic expressions

Open parenthesis ( indicates the start of a new level

Closing parenthesis ) completes the current level and moves up one level in the tree

A (B(C D) E F (G H I)) A (B(C D) E F (G H I))

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Binary trees

Binary tree:

 no node can have more than two subtrees.

 A node can have 0, 1, or 2 subtrees (L and R)

Null tree:

 tree with no nodes

Symmetry is not a tree requirement

AA

BB EE

CC DD FF

Right subtree Left subtree

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19

AA AA AA

BB BB

AA

BB CC

AA

BB CC

DD EE

AA

BB

CC

AA

BB

CC

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Height of Binary Trees

 Given N is the number of node in a binary tree

 Maximum Height: Hmax = N

 Minimum height: Hmin = [Log₂N]+1

 The shorter tree the easier it is to locate any desired node in the tree.

AA

BB

CC

AA

BB CC

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Number of Nodes in Binary Trees

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 Given H is the height of a binary tree

 Minimum number of nodes: Nmin = H and

 Maximum number of nodes: Nmax = 2^H- 1

AA

BB

CC

AA

BB CC

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22

Level Max nodes 0 1

1 2 2 4 3 8 4 16

…

N = 2 ^level

But: height = level +1 So: N = 2 ^height -1

Max height = [log₂N] +1

Binary trees

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Binary Trees

 Balance

 Balance factor:

 The difference in height between its left and right subtree

 Balanced tree

 Balance factor = 0 and its subtrees are also balanced

 Balanced tree in general

 If the height of its subtrees differs by no more than 1 (B= -1, 0, 1)

 and its subtrees are also balanced

B = H_L- H_R B = H_L- H_R

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AA AA AA

BB BB

AA

BB CC

AA

BB CC

DD EE

AA

BB

CC

AA

BB

CC

0 0 1 -1

0 1

-2 2

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Binary Trees

 Complete tree

 Has the maximum number of entries for its height

 The max num is reached when the last level is full

 N_max= 2^H - 1

 Nearly complete tree

 Has the minimum height of its nodes

 And all nodes in the last level are found in the left

 H_min = [log₂N]+1

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AA AA

BB CC

AA

BB CC

DD EE

AA

BB EE

CC DD FF GG

AA

BB CC

DD EE FF

AA

BB CC

DD

Nearly complete trees (at level 2) Complete trees (at levels 0, 1 and 2)

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Binary trees

 Each node in the structure must contain the data to be stored and two pointers, one in L subtree and one in the R subtree

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Binary tree traversal

 Requires that each node of the tree be processed once and only once in a predetermined sequence

 Depth-First traversal

» process all of the descendents of a child before going on to the next child.

 Breadth-First traversal

» Each level is completely processed before the next level is started

A

B E

C D F

A

B E

C D F

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Depth-First traversal

 Given a binary tree having a root, LST, RST:

 Define 6 DFT sequences

3 2

1

Preorder traversal NLR

L R

3 1

2

L R

2 1

3

L R

Inorder traversal LNR

Postorder traversal LRN

Standard Traversals

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Depth-First traversal



Preorder traversal: NLR

 root goes before the left and right subtrees

3 2

1

Preorder traversal NLR

L R

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44

3 1

2

L R

Inorder traversal LNR

 Inorder traversal: LNR

 (IN) root is processed in between subtrees

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Depth-First traversal

45

A

B E

C D F

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Depth-First traversal

46

A

B E

C D F

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Depth-First traversal

47

A

B E

C D F

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58

2 1

3

67

A

B E

C D F

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Introduction to Trees