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

2Universidade Federal da ParaíbaCentro de Informática

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


What is a Tree?


● Example:

3

5

6

4

1

2

4

1

2

3

4

5

Root node


What is a Tree?


● Example:

3

5

6

4

2

14

3

5 26

1

Root node


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


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


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


Definitions

● Subtree– Tree formed by a node

and all its descendants.

2

9

87

1

4

3

65


Definitions

● Tree Arity– n-ary.

– Binary. 2

9

87

4

3

65

1

n-ary tree

3

4

2

65

1

binary tree


Binary Tree Implementation

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

parent

value

left right

Node



parent

value

null

left null right null

Node

root

parent

value

null


Node

left child



parent

value

null

left right null

Node

root

parent

value


Node

How about a n-ary tree?


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.



● Example

Node

root

p

left rsib

n

n n



● Example

Node

root

p

left rsib

n

n

p

left rsibn n



● Example

Node

root

p

left rsib

n

n

p

left rsibn

p

left rsibn n



● Example

Node

root

p

left rsib

n

n

p

left rsibn

p

left rsibn

p

left rsibn n



● Example

Node

root

p

left rsib

n

n

p

left rsibn

p

left rsibn

p

left rsibn

p

left rsibn n



● 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



● 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


Tree Traversal

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

– Depth-first.


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




3

4

2

65

1



● 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:



● 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:



● 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:


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


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:


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:


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:

data structures - 13. introduction to trees

Education

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definitions tree arity

definitions subtree

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definitions child node