Data Structures and Algorithms: Tree Concepts and Implementations, Slides of Data Structures and Algorithms

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Data Structures &

Algorithm Analysis

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Definition of Tree

A tree is a finite set of one or more nodes

such that:

There is a specially designated node called

the root.

The remaining nodes are partitioned into n>=

disjoint sets T1, ..., Tn , where each of these sets is

a tree.

We call T1, ..., Tn the subtrees of the root.

Level and Depth

K L

E F

B

G

C

M

H I J

D

A

Level

1

2

3

4

node (13) degree of a node leaf (terminal) nonterminal parent children sibling degree of a tree (3) ancestor level of a node height of a tree (4)

Tree Properties

A

B C

D

G

E F

H I

Property Value Number of nodes Height Root Node Leaves Interior nodes Number of levels Ancestors of H Descendants of B Siblings of E Right subtree

Representation of Trees

List Representation

( A ( B ( E ( K, L ), F ), C ( G ), D ( H ( M ), I, J ) ) )

The root comes first, followed by a list of sub-trees

data link 1 link 2 ... link n

How many link fields are needed in such a representation?

Left Child - Right Sibling

A

B C^ D

E F^ G^ H^ I^ J

K L^ M

data left child right sibling

Tree ADT

• Objects: any type of objects can be stored in a tree
• Methods:
• accessor methods
  • root() – return the root of the tree
  • parent(p) – return the parent of a node
  • children(p) – returns the children of a node
• query methods
  • size() – returns the number of nodes in the tree
  • isEmpty() - returns true if the tree is empty
  • elements() – returns all elements
  • isRoot(p), isInternal(p), isExternal(p)

Tree Traversal

• Two main methods:

• Preorder
• Postorder

• Recursive definition

• PREorder:

• visit the root
• traverse in preorder the children (subtrees)

• POSTorder

• traverse in postorder the children (subtrees)

Preorder

• preorder traversal Algorithm preOrder(v) “visit” node v for each child w of v do recursively perform preOrder(w)

Preorder Implementation

public void preorder(ptnode t) {

ptnode ptr; display(t->key); for(ptr = t->lchild; NULL != ptr; ptr = ptr->sibling) { preorder(ptr); } }

Postorder Implementation

public void postorder(ptnode t) {

ptnode ptr; for(ptr = t->lchild; NULL != ptr; ptr = ptr->sibling) { postorder(ptr); } display(t->key); }

Example

J

M I

L^ H

A

B

C

D

E

K^ F^ G

ADT Binary Tree

objects: a finite set of nodes either empty or

consisting of a root node, left BinaryTree ,

and right BinaryTree.

method:

for all bt , bt1 , bt2 ∈ BinTree , item ∈ element

Bintree create()::= creates an empty binary tree

Boolean isEmpty( bt )::= if ( bt ==empty binary tree) return TRUE else return FALSE