introduction to c programming ce00312-1
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Introduction to C Programming CE00312-1. Lecture 23 Binary Trees. Binary Search Tree. A binary tree has a maximum of two branches. A binary search tree has some further characteristics: - PowerPoint PPT PresentationTRANSCRIPT
Introduction to C Programming
CE00312-1
Lecture 23
Binary Trees
Binary Search Tree
A binary tree has a maximum of two branches.
A binary search tree has some further characteristics:
1) With respect to any particular node in the tree, all those nodes in the left sub-tree have data which are less (alphabetically for strings) than the node, and all those in the right sub-tree are greater.
A Binary Search Tree
teddy
fred
nickdarryl
fong
thomas
rob
brian
colin
a leaf
root of tree
Searching a Binary Search Tree 2) A search for a particular data item, only involves
searching down one branch at each node.
In the example, searching for “fong” involves
going left at “fred” (“fong” < “fred”), then right at “colin” (“fong” > “colin”), then right at “darryl” (“fong” > “darryl”) and then finding “fong” as a leaf.
Inserting into a Binary Search Tree
teddy
fred
nickdarryl
fong
thomas
rob
brian
colin
claudenode toinsert
a leaf
root of tree
Insertion into a Binary Search Tree
3) Insertion of a new node is similar to a search, and then linking in a new leaf for that node.
In the example, a new node for “claude” would be inserted to the right of “brian”, because “claude” is less than both “fred” and “colin” but not “brian”.
4) An Inorder Traversal of the tree yields all the nodes in order (alphabetical for strings).
Efficiency Searching, insertion, deletion and sorting (see below)
are efficient because half the tree is eliminated at each comparison (cf binary search with arrays).
In searching for an item in a binary search tree only involves going left or right for each node as we descend the tree. This is similar to choosing first or second half during a binary search for an array.
Eliminating half the data with each comparison implies the total number of comparisons is log2 n for n items.
However, this is only guaranteed if the tree is balanced!
A Binary Tree as a Linked List
fred
teddy
nick
NULL
NULL
NULL
NULL
NULL
root
darryl
Structure of a tree node
left rightdata
left sub-tree right sub-tree
Header for Binary Search Tree Let us define a header file called “tree.h” containing all
our types, structures and tree function prototypes. This file can be included in all files that need to use
binary search trees.
We shall also keep all our tree functions in “tree.c” and an application in “treegrow.c”.
All these may be compiled by:cc treegrow.c tree.c
Header file “tree.h”
#include "stdio.h“ // for I/O and NULL#include "stdlib.h“ // for malloc#include "string.h" // for strings
struct node{ struct node *left; // left branch
char data[21]; // string data struct node *right; // right branch};typedef struct node *Treepointer; // new type called Treepointer
Prototypes for binary search trees
void inorder(Treepointer);// traverse tree
void insert(Treepointer, Treepointer);// insert a node into a tree
Treepointer createnode(char []);
// create a node for an item
Treepointer delete(Treepointer, char []);// delete an item from a tree
Inorder Traversal
A traversal involves visiting all the nodes in the tree in a particular sequence.
The most commonly used one is the inorder traversal. Inorder traversal
1. visits all the nodes to left of the given node,
2. then the given node itself and
3. then visits all those to the right.
For a binary search tree this yields the data in
sort order.
Traversing a binary search tree
teddy
fred
nickdarryl
fong
thomas
rob
brian
colin
a leaf
root of tree
Traversals are recursive Any function that visits all the nodes in a tree has to be
recursive.
All traversal algorithms should be recursive.
Iterative (using while loops) solutions are extremely cumbersome - not to mention very difficult - to write.
This should be expected because trees themselves are
recursive data structures - every tree has branches which are themselves subtrees.
Inorder Traversal
#include "tree.h"void inorder(Treepointer T){ // traverse the tree, T if (T != NULL)
{ inorder(T -> left); // traverse left
printf("%s\n", T -> data);// print data
inorder(T -> right); // traverse right } // else empty tree do nothing}
To print all nodes in alphabetical order use:inorder (root);
Algebraic Trees
Inorder Traversal can give the same result for different trees.
Using the normal infix notation, brackets would have to be used to distinguish
(A + B) * C from A + B * CE.G.
(2 + 3) * 4 2 + 3 * 4Give
20 14
Reverse Polish However, postorder traversal gives different results
for different trees.
Thus, reverse polish notation can represent algebraic trees faithfully without the use of either brackets or operator precedences!
A B + C * (do A B + first)is different from
A B C * + (do B C * first)
Hence the use of reverse polish by compilers
Postorder traversal
Postorder traversal is almost the same as inorder traversal – both have to visit all the same nodes – but in a different sequence.
The recursive algorithm should be very similar as the two traversals do similar things.
Only two statements are swapped, so that for a postorder traversal, the right subtree is visited before printing the current node, T.
Postorder Traversal
#include "tree.h"void postorder(Treepointer T){ // traverse the tree, T if (T != NULL)
{ postorder(T -> left); // traverse left
postorder(T -> right);// traverse right printf("%s ", T -> data);// print data
} // else empty tree do nothing}
To print all nodes in reverse polish use:postorder (root);