Chapter 20 Binary Trees

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Copyright © 2012 Pearson Education, Inc.Copyright © 2012 Pearson Education, Inc.

Chapter 20:

Binary Trees




20.1Definition and Application of

Binary Trees




Definition and Application of

Binary Trees• Binary tree: a nonlinear linked list in which each

node may point to 0, 1, or two other nodes

• Each node contains

one or more

data fields and

two pointers

NULL NULL

NULL NULL NULL NULL




Binary Tree Terminoloy

• Tree pointer: like a

head pointer for a

linked list, it points to

the first node in the!inary tree

• "oot node: the node at

the top of the tree

NULL NULL

NULL NULL NULL NULL





• #eaf nodes: nodes thatha$e no children

The nodes containin7 and 43 are leafnodes

NULL NULL7

19

31

43

59

NULL NULL NULL NULL





• %hild nodes,children: nodes!elow a i$en node

The children of thenode containin 31 are the nodescontainin 19 and59

NULL NULL7

19

31

43

59

NULL NULL NULL NULL


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• &arent node: nodea!o$e a i$en node

The parent of the nodecontainin 43 is thenode containin 59

NULL NULL7

19

31

43

59

NULL NULL NULL NULL




• '(!tree: the portion of

a tree from a node

down to the lea$es

The nodes containin

19 and 7 are the left

s(!tree of the node

containin 31 NULL NULL7

19

31

43

59

NULL NULL NULL NULL



)ses of Binary Trees

• Binary search tree: dataorani*ed in a !inary treeto simplify searches

• #eft s(!tree of a node

contains data $al(es +the data in the node

• "iht s(!tree of a nodecontains $al(es the

data in the nodeNULL NULL7

19

31

43

59

NULL NULL NULL NULL



'earchin in a Binary Tree

1- 'tart at root node

2- Eamine node data:a- /s it desired $al(e Done

!- Else, is desired data + node

data "epeat step 2 with lefts(!tree

c- Else, is desired data nodedata "epeat step 2 withriht s(!tree

- %ontin(e (ntil desired$al(e fo(nd or NULL pointer reached

NULL NULL7

19

31

43

59

NULL NULL NULL NULL



'earchin in a Binary TreeTo locate the node containin 43,

Eamine the root node 331- first

'ince 43 > 31, eamine the riht child of the node containin 31, 359-

'ince 43 < 59, eamine the left child of the node containin 59, 343-

The node containin4 has !een fo(nd

NULL NULL7

19

31

43

59

NULL NULL NULL NULL



20.2Binary 'earch Tree 5perations



Binary 'earch Tree 5perations

• %reate a !inary search tree orani*e datainto a !inary search tree

• /nsert a node into a !inary tree p(t node intotree in its correct position to maintain order

• 6ind a node in a !inary tree locate a nodewith partic(lar data $al(e

• Delete a node from a !inary tree remo$e anode and ad7(st links to maintain !inary tree



Binary 'earch Tree 8ode

• A node in a !inary tree is like a node ina linked list, with two node pointer fields:

struct TreeNode

{

int value;

TreeNode *left;

TreeNode *rigt;!




%reatin a 8ew 8ode

• Allocate memory for new node:ne"Node # ne" TreeNode;

• /nitiali*e the contents of the node:

ne"Node$>value # nu%;• 'et the pointers to NULL:

ne"Node$>Left

# ne"Node$>&igt

# NULL; ne"Node

'3

NULLNULL

ne"Node

ne"Node

'3




/nsertin a 8ode in a

Binary 'earch Tree

1- /f tree is empty, insert the new node as the rootnode

2- Else, compare new node aainst left or rihtchild, dependin on whether data $al(e of newnode is < or > root node

- %ontin(e comparin and choosin left or rihts(!tree (ntil NULL pointer fo(nd

4- 'et this NULL pointer to point to new node




/nsertin a 8ode in a

Binary 'earch Tree

NULL NULL7

19

31

43

59

root

Eamine this node first $al(e is + node, so o toleft s(!tree

Eamine this nodesecond $al(e is node, so o toriht s(!tree

'ince the riht s(!treeis NULL, insert here

NULL NULL NULL NULL

ne"Node

'3

NULLNULL




Tra$ersin a Binary Tree

Three tra$ersal methods:1- /norder:

a- Tra$erse left s(!tree of node

!- &rocess data in node

c- Tra$erse riht s(!tree of node2- &reorder:

a- &rocess data in node

!- Tra$erse left s(!tree of node

c- Tra$erse riht s(!tree of node

- &ostorder:a- Tra$erse left s(!tree of node

!- Tra$erse riht s(!tree of node

c- &rocess data in node




Tra$ersin a Binary Tree

T"A9E"'A#ET;5D

85DE'9/'/TED /85"DE"

/norder 7( 19( 31(

43( 59

&reorder 31( 19( 7(

59( 43

&ostorder 7( 19( 43(

59( 31

NULL NULL7

19

31

43

59

NULL NULL NULL NULL




'earchin in a Binary Tree

• 'tart at root node,

tra$erse the tree

lookin for $al(e

• 'top when $al(efo(nd or NULL

pointer detected

• %an !e implemented

as a )ool f(nction

NULL NULL7

19

31

43

59

NULL NULL NULL NULL

'earch for 43 ret(rn true'earch for 17 ret(rn false




Deletin a 8ode from a

Binary Tree #eaf 8ode• /f node to !e deleted is a leaf node, replace

parent node<s pointer to it with a NULL pointer,

then delete the node

NULL7

19

NULL NULL

Deletin node with = !efore deletion

NULL

19

NULL

Deletin node with = after deletion





Binary Tree 5ne %hild

• /f node to !e deleted has one child node,

ad7(st pointers so that parent of node to !e

deleted points to child of node to !e deleted,

then delete the node





Binary Tree 5ne %hild

NULL NULL7

19

31

43

59

NULL NULL NULL NULL

Deletin node with 1> !efore deletion

NULL

7

31

43

59

NULL NULL

NULL NULL

Deletin node with 1> after deletion





Binary Tree Two %hildren

• /f node to !e deleted has left and riht children, ?&romote< one child to take the place of the deleted

node

#ocate correct position for other child in s(!tree ofpromoted child

• %on$ention in tet: promote the riht child,position left s(!tree (nderneath





Binary Tree Two %hildren

NULL NULL7

19

31

43

59

NULL NULL NULL NULL

Deletin node with 1 !efore deletion Deletin node with 1

after deletion

43

59

NULL

NULL

7

19

NULL NULL

NULL




20.Template %onsiderations for

Binary 'earch Trees

% f



C i ht © 2012 P Ed ti IC i ht © 2012 P Ed ti I

Template %onsiderations for

Binary 'earch Trees

• Binary tree can !e implemented as a

template, allowin flei!ility in determinin

type of data stored• /mplementation m(st s(pport relational

operators >, <, and ## to allow

comparison of nodes

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Chapter 20 Binary Trees