Threaded Binary Tree

Definition

A binary search tree in which each node uses an otherwise-empty left child link to refer to the node's in-order predecessor and an empty right child link to refer to its in-order successor.

Threaded Binary Trees

• Two many null pointers in current representationof binary trees

n: number of nodesnumber of non-null links: n-1total links: 2nnull links: 2n-(n-1)=n+1

• Replace these null pointers with some useful“threads”.

Threaded Binary Trees (Continued)

If ptr->left_child is null,

replace it with a pointer to the node that would be

visited before ptr in an inorder traversal

If ptr->right_child is null,

replace it with a pointer to the node that would be

visited after ptr in an inorder traversal

A Threaded Binary Tree

A

B C

GE

I

D

H

F

root

dangling

dangling

inorder traversal:

H, D, I, B, E, A, F, C, G

TRUE � � FALSE

Data Structures for Threaded BT

typedef struct threaded_tree

*threaded_pointer;

typedef struct threaded_tree {

short int left_thread;

threaded_pointer left_child;

char data;

threaded_pointer right_child;

short int right_thread; };

left_thread left_child data right_child right_thread

FALSE: childTRUE: thread

• Dangling can be solved as follows

– Introduce a header node.

– The left and right pointer of the header node

are treated as normal links and are initialized

to point to header node itself

Memory Representation of A Threaded BT

f f--

f fA

f fCf fB

t tE t tF t Gf fD

t tIt tH

root

• Right-threaded tree:

• A variant of a threaded tree in which only the right thread, i.e. link to the successor, of each node is maintained.

Advantages of Threaded Tree

• It reduces the time complexity of algorithm.

• Operation like traversal, insertion, deletion can be done without using recursive algorithm.

• It becomes very easy to find out inorder successor of any node ‘n’ in a threaded binary tree. If the right thread of n is true or 1 then the node pointed by its right pointer is the desired information successor. If the right thread of ‘n’ is false or 0 then the inorder successor must be on the right subtree of ‘n’.

Application of Tree

• Expression Tree

• Game Tree

Infix, prefix, and postfix notation

• representation of math expressions as a binary tree

– operators have their left and right operands as

subtrees

– literal values are stored as leaves

• notations

– prefix: Polish notation

– infix: standard notation

– postfix: reverse Polish notation

Expression example

Expression tree example

Game Tree• Tree generated for games such as tic-tac-

toe

• Consider all the possible position for the given position. And find the best solution to win.

Heap Sort

• It is based on tree structure.

• Heap sort is in two phase

– The entries are arranged in the list satisfying

the requirement of a heap

– We repeatedly remove the top of the heap

and promote another entry to take its place.

Construction of heap

• First construct a complete binary tree

• There are two types of heap- min heap or ascending heap or max heap or descending heap

• Consider the elements present at a level one less than the maximum level. They are converted to heap in the same way as replacing the node of a heap.

• In next step the elements that are present at a level two less than the maximum level of the tree are considered

• Likewise in each step one level is decremented and all the subtrees at that level are converted to heaps

• Eg 13, 4, 11, 15, 59, 27, 19, 3, 92, 5

Phase two

• Remove the root of tree and place it to last and re-create the heap in descending order and repeat the process.

• Solution on board.