lec 9 trees

7/23/2019 Lec 9 Trees

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1

Trees

Lecture # 9

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Tree

A tree is a collection of n nodes, one of which is the root,and n -1 edges.

n -1 edges, because each edge connects some node to itsparent, and every node except the root has one parent

ach node may have an arbitrary number of children,possibly !ero. "odes with no children are nown as leaves."odes with the same parent are siblings

A Path from node n1 to n is defined as a se$uence of

nodes n1, n%, &,n such that ni is the parent of ni ' 1 for 1 ≤ i ( .

)he length of this path is the number of edges on the path,namely * 1

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Tree

)here is a path of length !ero from every

node to itself.

+n a tree there is exactly one path fromthe root to each node.

)he level of any node is the length of its

path from the root.

)he degree of a node is its number of

subtrees.

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Binary Trees

A binary tree is either empty, or it consists of a node calledthe root together with two binary trees called the left sub-tree and right sub-tree of the root

A binary tree consists of a root and one or two children named as left child and right child.

A binary tree is a set of elements that is either empty or ispartitioned into three disjoint subsets.

)he first subset contains a single element called the root ofthe tree. )he other two subsets are themselves binary trees,called the left and right sub-trees of the original tree.

A left or right sub-tree can be empty. ach element of the binary tree is called the node of the

tree.

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Binary Trees

Some Binary Trees

One node Two nodes

Three nodes

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Binary Trees

trictly inary )ree +f every non-leaf node in a binary tree has non-empty

left and right sub-trees A strict binary tree with n

leaves always contain 2n-1 nodes. omplete inary )ree

A strictly binary tree all of whose leaves are at level d

+f a binary tree contains m nodes at levell

it containsat most %m nodes at level l '1.

A complete binary tree contains 2l nodes at level l .

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Dynamic Implementation of

Binary Tree Linked Implementation

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Binary Trees Implementation

class BTNode

{

public

in! in"o#

BTNode $le"!% $ri&h!#

BTNode'( { le"! ) ri&h! ) *# +

BTNode'in! el% BTNode $le" ) *% BTNode $re! ) *( {

in"o ) el#

le"! ) le"#ri&h! ) re!#

+

+#

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,


class BT

{

public

BT' ( { roo! ) *# +

-BT' ( { clear' (# roo! ) *# +.oid clear' ( { clear'roo!(# roo! ) *# + //0ele!e all nodes o" a !ree

bool is1mp!y' ( { re!urn roo! )) * +//checs !ree is emp!y or no!

.oid preorder' ( { preorder'roo!(# +//preorder !ra.ersal o" !ree

.oid inorder' ( { inorder'roo!(# +//inorder !ra.ersal o" !ree

.oid pos!order' ( { pos!order'roo!(# +//pos!order !ra.eral o" !ree

.oid bread!hirs!'(#//bread!h "irs! !ra.ersal

on!inue on ne! slide

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*


pri.a!e

BTNode $roo!#

.oid clear'BTNode $(#

.oid preorder'BTNode $(#

.oid inorder'BTNode $(#

.oid pos!order'BTNode $(#

+#

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Binary Trees Traversals

)ree )raversal is the process of visiting each

node in the tree exactly one time.

)ree traversals are of two types /epth 0irst )raversal

readth 0irst )raversal

)he three /epth 0irst )raversal techni$ues are

reorder tree traversal +norder tree traversal

ostorder tree traversal

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reorder )ree )raversal

1. 2isit the root

%. )raverse the left sub-tree in preorder 3. )raverse the right sub-tree in preorder

Example:

AB

C

ED9B01

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.oid BTpreorder'BTNode $p(

{

i"'p :) *(

{

cou!;;p<=in"o#

preorder'p<=le"!(#

preorder'p<=ri&h!(#

+

+

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+norder )ree )raversal4

1. )raverse the left sub-tree in inorder

%. 2isit the root3. )raverse the right sub-tree in inorder

Example:

A

BC

ED901B

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.oid BTinorder'BTNode $p(

{

i"'p :) *(

{

inorder'p<=le"!(#

cou!;;p<=in"o#

inorder'p<=ri&h!(#

+

+

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ostorder )ree )raversal4

1. )raverse the left sub-tree in postorder

%. )raverse the right sub-tree in postorder 3. 2isit the root

Example:

A

BC

ED01B9

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.oid BTpos!order'BTNode $p(

{

i"'p :) *(

{

pos!order'p<=le"!(#

pos!order'p<=ri&h!(#

cou!;;p<=in"o#

+

+

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Traverse the following tree.

+

- /

* ^ E F

A B C D

>norder 9 $ B ? @ 0 A 1 /

reorder A * $ 9 B @ 0 / 1

os!order 9 B $ 0 @ < 1 / A

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,


11!

"# 1$

%&1'

1(

1

"(

!

!

)re*order Traversal+

)ost*order Traversal+

In*order Traversal+

Traverse the following tree.

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2*


Breadth First Tree Traversal +mplementation of this ind of traversal is

straightforward when a ueue is used.

onsider a top down left-to-right, breadth

first traversal.

After a node is visited, its children, if any

are placed at the end 5rear6 of a $ueue, andthe node at the beginning 5front6 of the

$ueue is visited.

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Breadth First TraversalExample

A

C

D

B

E

,-

I

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C

/0e0e

B

A

C

D

B

E

,-

I

9

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D0e0e C

B

A

C

D

B

E

,-

I

9


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B

En0e0 D

D

A

C

D

B

E

,-

I

9

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D0e0e B

D

A

C

D

B

E

,-

I

9B

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D

En0e0e E2

E

A

C

D

B

E

,-

I

9B

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D0e0e D

E

A

C

D

B

E

,-

I

9B0

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D0e0e E

A

C

D

B

E

,-

I

9B01

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En0e0e ,2 -

, -

A

C

D

B

E

,-

I

9B01

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D0e0e

, -

A

C

D

B

E

,-

I

9B01C

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I

En0e0e I

, -

A

C

D

B

E

,-

I

9B01C

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A

C

D

B

E

,-

I

I

D0e0e ,

-

9B01C

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A

C

D

B

E

,-

I

I

D0e0e -

9B01CD

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A

C

D

B

E

,-

I

D0e0e I

9B01CD>

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Binary Trees Traversals .oid BTbread!hirs!'(

{ Eueue Fueue#

BTNode $p ) roo!#

i"'p :) *(

{ FueueenFueue'p(#

while':Fueueemp!y'((

{ p ) FueuedeFueue'(#cou!;;p<=ey#

i"'p <= le"! :) *( FueueenFueue'p<=le"!(#

i"'p <= ri&h! :) *( FueueenFueue'p<=ri&h!(#

+

++

lec 9 trees

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