2-3 Trees | (Search, Insert and Deletion)

In  binary search trees we have seen the average-case time for operations like search/insert/delete is O(log N) and the worst-case time is O(N) where N is the number of nodes in the tree.

Like other Trees include AVL trees, Red Black Tree, B tree, 2-3 Tree is also a height balanced tree.

The time complexity of search/insert/delete is O(log N) .

A 2-3 tree is a B-tree of order 3.

Properties of 2-3 tree:

• Nodes with two children are called 2-nodes. The 2-nodes have one data value and two children
• Nodes with three children are called 3-nodes. The 3-nodes have two data values and three children.
• Data is stored in sorted order.
• It is a balanced tree.
• All the leaf nodes are at same level.
• Each node can either be leaf, 2 node, or 3 node.
• Always insertion is done at leaf.

Search: To search a key K in given 2-3 tree T, we follow the following procedure: 

Base cases: 

1. If T is empty, return False (key cannot be found in the tree).
2. If current node contains data value which is equal to K, return True.
3. If we reach the leaf-node and it doesn't contain the required key value K, return False.

Recursive Calls: 

1. If K < currentNode.leftVal, we explore the left subtree of the current node.
2. Else if currentNode.leftVal < K < currentNode.rightVal, we explore the middle subtree of the current node.
3. Else if K > currentNode.rightVal, we explore the right subtree of the current node.

Consider the following example:

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Insertion: There are 3 possible cases in insertion which have been discussed below: 

• Case 1: Insert in a node with only one data element 

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• Case 2: Insert in a node with two data elements whose parent contains only one data element. 
   

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• Case 3: Insert in a node with two data elements whose parent also contains two data elements.  

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In Deletion Process for a specific value:

• To delete a value, it is replaced by its in-order successor and then removed.
• If a node is left with less than one data value then two nodes must be merged together.
• If a node becomes empty after deleting a value, it is then merged with another node.

To Understand the deletion process-

Consider the 2-3 tree given below

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delete the following values from it: 69,72, 99, 81.

To delete 69, swap it with its in-order successor, that is, 72. 69 now comes in the leaf node. Remove the value 69 from the leaf node.

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To delete 72, 72 is an internal node. To delete this value swap 72 with its in-order successor 81 so that 72 now becomes a leaf node. Remove the value 72 from the leaf node.

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Now there is a leaf node that has less than 1 data value thereby violating the property of a 2-3 tree. So the node must be merged.

To merge the node, pull down the lowest data value in the parent’s node and merge it with its left sibling.

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To delete 99, 99 is present in a leaf node, so the data value can be easily removed.

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Now there is a leaf node that has less than 1 data value, thereby violating the property of a 2-3 tree.

So the node must be merged. To merge the node, pull down the lowest data value in the parent’s node and merge it with its left sibling.

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To delete 81, 81 is an internal node. To delete this value swap 81 with its in-order successor 90 so that 81 now becomes a leaf node. Remove the value 81 from the leaf node.

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Now there is a leaf node that has less than 1 data value, thereby violating the property of a 2-3 tree. So the node must be merged. To merge the node, pull down the lowest data value in the parent’s node and merge it with its left sibling.

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As internal node cannot be empty. So now pull down the lowest data value from the parent’s node and merge the empty node with its left sibling

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NOTE: In a 2-3 tree, each interior node has either two or three children. This means that a 2-3 tree is not a binary tree.