In computer science, a binary search tree (BST) is a node based binary tree data structure which has the following properties:
- The left subtree of a node contains only nodes with keys less than the node's key.
- The right subtree of a node contains only nodes with keys greater than the node's key.
- Both the left and right subtrees must also be binary search trees.
From the above properties it naturally follows that:
- Each node (item in the tree) has a distinct key.
Generally, the information represented by each node is a record rather than a single data element. However, for sequencing purposes, nodes are compared according to their keys rather than any part of their their associated records.
The major advantage of binary search trees over other data structures is that the related sorting algorithms and search algorithms such as in-order traversal can be very efficient.
Binary search trees are a fundamental data structure used to construct more abstract data structures such as sets, multisets, and associative arrays.
A binary search tree of size 9 and depth 3, with root 8 and leaves 1, 4, 7 and 13
typedef int ElementType;
#ifndef _Tree_H
#define _Tree_H
struct TreeNode;
typedef struct TreeNode *Position;
typedef struct TreeNode *SearchTree;
SearchTree MakeEmpty( SearchTree T );
Position Find( ElementType X, SearchTree T );
Position FindMin( SearchTree T );
Position FindMax( SearchTree T );
SearchTree Insert( ElementType X, SearchTree T );
SearchTree Delete( ElementType X, SearchTree T );
ElementType Retrieve( Position P );
#endif /* _Tree_H */
#include "tree.h"
#include <stdlib.h>
#include "fatal.h"
struct TreeNode
{
ElementType Element;
SearchTree Left;
SearchTree Right;
};
SearchTree MakeEmpty( SearchTree T )
{
if( T != NULL )
{
MakeEmpty( T->Left );
MakeEmpty( T->Right );
free( T );
}
return NULL;
}
Position Find( ElementType X, SearchTree T )
{
if( T == NULL )
return NULL;
if( X < T->Element )
return Find( X, T->Left );
else
if( X > T->Element )
return Find( X, T->Right );
else
return T;
}
Position FindMin( SearchTree T )
{
if( T == NULL )
return NULL;
else
if( T->Left == NULL )
return T;
else
return FindMin( T->Left );
}
Position FindMax( SearchTree T )
{
if( T != NULL )
while( T->Right != NULL )
T = T->Right;
return T;
}
SearchTree Insert( ElementType X, SearchTree T )
{
/* 1*/ if( T == NULL )
{
/* Create and return a one-node tree */
/* 2*/ T = malloc( sizeof( struct TreeNode ) );
/* 3*/ if( T == NULL )
/* 4*/ FatalError( "Out of space!!!" );
else
{
/* 5*/ T->Element = X;
/* 6*/ T->Left = T->Right = NULL;
}
}
else
/* 7*/ if( X < T->Element )
/* 8*/ T->Left = Insert( X, T->Left );
else
/* 9*/ if( X > T->Element )
/*10*/ T->Right = Insert( X, T->Right );
/* Else X is in the tree already; we'll do nothing */
/*11*/ return T; /* Do not forget this line!! */
}
SearchTree Delete( ElementType X, SearchTree T )
{
Position TmpCell;
if( T == NULL )
Error( "Element not found" );
else
if( X < T->Element ) /* Go left */
T->Left = Delete( X, T->Left );
else
if( X > T->Element ) /* Go right */
T->Right = Delete( X, T->Right );
else /* Found element to be deleted */
if( T->Left && T->Right ) /* Two children */
{
/* Replace with smallest in right subtree */
TmpCell = FindMin( T->Right );
T->Element = TmpCell->Element;
T->Right = Delete( T->Element, T->Right );
}
else /* One or zero children */
{
TmpCell = T;
if( T->Left == NULL ) /* Also handles 0 children */
T = T->Right;
else if( T->Right == NULL )
T = T->Left;
free( TmpCell );
}
return T;
}
ElementType Retrieve( Position P )
{
return P->Element;
}
#include "tree.h"
#include <stdio.h>
main( )
{
SearchTree T;
Position P;
int i;
int j = 0;
T = MakeEmpty( NULL );
for( i = 0; i < 50; i++, j = ( j + 7 ) % 50 )
T = Insert( j, T );
for( i = 0; i < 50; i++ )
if( ( P = Find( i, T ) ) == NULL || Retrieve( P ) != i )
printf( "Error at %d\n", i );
for( i = 0; i < 50; i += 2 )
T = Delete( i, T );
for( i = 1; i < 50; i += 2 )
if( ( P = Find( i, T ) ) == NULL || Retrieve( P ) != i )
printf( "Error at %d\n", i );
for( i = 0; i < 50; i += 2 )
if( ( P = Find( i, T ) ) != NULL )
printf( "Error at %d\n", i );
printf( "Min is %d, Max is %d\n", Retrieve( FindMin( T ) ),
Retrieve( FindMax( T ) ) );
return 0;
}