data structures - 13. introduction to trees

Universidade Federal da ParaíbaCentro de Informática

Introduction to TreesLecture 16

1107186 – Estrutura de Dados – Turma 02

Prof. Christian Azambuja PagotCI / UFPB

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What is a Tree?

● In graph theory, a tree is defined as a connected acyclic graph.

● Example:

3

5

6

4

1

2 However, one such graph may generate

different tree data structures!

Node

Edge

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What is a Tree?

● In graph theory, a tree is defined as a connected acyclic graph.

● Example:

3

5

6

4

1

2

4

1

2

3

4

5

Root node

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What is a Tree?

● In graph theory, a tree is defined as a connected acyclic graph.

● Example:

3

5

6

4

2

14

3

5 26

1

Root node

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Definitions

● Path Length– Number of edges on a path.

● Depth of a Node– Equal to the path length to the root.

● Height of a Node– Equal to the path length from the

node to its deepest descendant.

● Height of a Tree– Equal to the height of the root.

2

9

87

1

4

3

65

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Definitions

● Child Node– Each node can have n

child nodes.

● Parent Node– It is unique for each

node.

– The root does not have parent node.

2

9

87

1

4

3

65

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Definitions

● Leaf Node– Has no child node.

● Siblings– Nodes with the same parent.

● Ancestors– All nodes on the path to the root.

● Descendants of a node n– All nodes that share n as an

ancestor.

2

9

87

1

4

3

65

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Definitions

● Subtree– Tree formed by a node

and all its descendants.

2

9

87

1

4

3

65

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Definitions

● Tree Arity– n-ary.

– Binary. 2

9

87

4

3

65

1

n-ary tree

3

4

2

65

1

binary tree

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Binary Tree Implementation

● Node– Pointers to parent, left and right child.

parent

value

left right

Node

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Binary Tree Implementation

parent

value

null

left null right null

Node

root

parent

value

null

left null right null

Node

left child

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Binary Tree Implementation

parent

value

null

left right null

Node

root

parent

value

left null right null

Node

How about a n-ary tree?

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n-ary Tree Implementation

● Node– Pointers to parent, left child and right sibling.

parent

value

left rsibling

Node

According to “Introduction to Algorithms”, Cormen, Leiserson, Rivest, Stein.

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n-ary Tree Implementation

● Example

Node

root

p

left rsib

n

n n

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n-ary Tree Implementation

● Example

Node

root

p

left rsib

n

n

p

left rsibn n

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n-ary Tree Implementation

● Example

Node

root

p

left rsib

n

n

p

left rsibn

p

left rsibn n

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n-ary Tree Implementation

● Example

Node

root

p

left rsib

n

n

p

left rsibn

p

left rsibn

p

left rsibn n

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n-ary Tree Implementation

● Example

Node

root

p

left rsib

n

n

p

left rsibn

p

left rsibn

p

left rsibn

p

left rsibn n

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n-ary Tree Implementation

● Example

Node

root

p

left rsib

n

n

p

left rsibn

p

left rsibn

p

left rsib

p

left rsibn n

p

left rsibn n

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n-ary Tree Implementation

● Example

Node

root

p

left rsib

n

n

p

left rsibn

p

left rsibn

p

left rsib

p

left rsibn n

p

left rsibn

p

left rsibn n

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Tree Traversal

● There are two types of tree traversal algorithms:– Breadth-first.

– Depth-first.

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Tree Traversal: Breadth-first-search (BFS)

● Visits only once each node of a graph on a level-by-level basis.

3

4

2

65

1

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Tree Traversal: Breadth-first-search (BFS)

● Visits only once each node of a graph on a level-by-level basis.

3

4

2

65

1

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Tree Traversal: Breadth-first-search (BFS)

● Visits only once each node of a graph on a level-by-level basis.

3

4

2

65

1

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Tree Traversal: Breadth-first-search (BFS)

● Visits only once each node of a graph on a level-by-level basis.

3

4

2

65

1

26Universidade Federal da ParaíbaCentro de Informática

Tree Traversal: Breadth-first-search (BFS)

● Visits only once each node of a graph on a level-by-level basis.

3

4

2

65

1

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Tree Traversal: Breadth-first-search (BFS)

● Implementationstruct Node{

struct Node* parent;int value;struct Node* left;struct Node* right;

};

struct Queue{

int first;int last;struct Node* elements[10];

};

struct Queue q;... int main(...) ...

C code excerpt:

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Tree Traversal: Breadth-first-search (BFS)

● Implementationvoid Enqueue(struct Queue* q, struct Node* n){

q­>elements[++q­>last] = n;}

struct Node* Dequeue(struct Queue* q){

return q­>elements[q­>first++];}

int QueueNotEmpty(struct Queue* q){

return q­>first <= q­>last;}

... int main(...) ...

C code excerpt:

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Tree Traversal: Breadth-first-search (BFS)

● Implementationvoid BFS(struct Node* n){

if (n != NULL){

Enqueue(&q,n);

while(QueueNotEmpty(&q)){

struct Node* v = Dequeue(&q);

printf("node value: %i\n",v­>value);

if (v­>left != NULL)Enqueue(&q,v­>left);

if (v­>right != NULL)Enqueue(&q,v­>right);

}}

}... int main(...) ...

C code excerpt:

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Tree Traversal: Depth-First-Search (DFS)

● Explores as far as possible each branch of the tree before backtracking.– Pre-order:

– In-order:

– Post-order:3

4

2

65

11, 2, 4, 5, 6, 3

5, 4, 6, 2, 1, 3

5, 6, 4, 2, 3, 1

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Tree Traversal: Pre-Order

● Implementation

void PreOrder(struct Node* n){

if (n != NULL){

printf("node value: %i\n", n­>value);PreOrder(n­>left);PreOrder(n­>right);

}}

C code excerpt:

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Tree Traversal: In-Order

● Implementation

void InOrder(struct Node* n){

if (n != NULL){

InOrder(n­>left);printf("node value: %i\n", n­>value);InOrder(n­>right);

}}

C code excerpt:

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Tree Traversal: Post-Order

● Implementation

void PostOrder(struct Node* n){

if (n != NULL){

PostOrder(n­>left);PostOrder(n­>right);printf("node value: %i\n", n­>value);

}}

C code excerpt: