BST with duplicates

问题:

I know that, BST does not allow duplicates. For example, if I have a word "RABSAB".

The Binary search tree for the above string is:

    R
    /\
   A  S
    \
     B

What if we wanted to include the duplicates in the tree. How the tree gonna change? I was asked this question in an interview.

They asked me to draw:

1. a binary tree
2. an unbalanced Binary Search Tree
3. a binary search tree without duplicates
4. a binary search tree with duplicates

Any Help is appreciated!

PS: Help me by drawing the related trees

回答1:

Rule to insert in a binary Search tree without duplicate is:

1. Go left if element is less than root
2. Go right if the element is greater than root.

And to allow duplicate entries you have to modify the rule like bellow:

1. Go left if the element is less or equal root
2. Go right if the element is greater than root.

or

1. Go left if the element is less than root
2. Go right if the element is greater or equal root.

or

1. Go left if the element is less than root
2. Go right if the element is greater than root.
3. Increase the count if the element is equal to the root.

So your BST for word "RABSAB", with duplicates can be like:

     R
    / \
   A   S
  / \
 A   B
    /
   B

Or,

     R
    / \
   A   S
    \
     A
      \
       B
        \
         B

or

    R(1)
   /  \
  /    \
 A(2)  S(1)
  \
   \
   B(2)

In First two cases, both insertion and search becomes bit complex! You will find it here with great deal of explanation!

And the third case is somewhat easier to maintain.

All of them are used successfully to allow duplicates, now the choice is yours!

回答2:

One option is to modify the tree so that one branch will include the duplicates, for example have the left branches hold nodes that are less than or equal to the parent, alternatively have the right branches hold nodes that are greater than or equal to the parent

Another option is to store all duplicates in a node, so instead of

class Node {
    Node left, right;
    Object data;
}

you would instead have

class Node {
    Node left, right;
    List data;
}

or

class Node {
    Node left, right;
    Object data;
    int count;
}

回答3:

In a normal BST insertion and search both occur based on less than(>) and greater than (<) rule.

You could instead try insertion on less than equal to (>=) or greater than equal to (<=) and try using the same rule for searching.

Alternatively you could include an array in every node to accommodate duplicate elements.

回答4:

For your input RABPAB, You can create a BST by using a LIST to store all the equal valued keys. All the equal valued keys are placed in the same level using a data structure capable of storing it.

The BST will look something like this,

     R
    / \
A--A   P
    \
  B--B

The Java code for your BST storing integer values could be like this,

class Node 
{
    Node left, right;
    int data[maxvalue];
}

Here maxvalue is the maximum possible equal valued keys.