# 2-3 Trees | (Search and Insert)

2-3 tree is a tree data structure in which every internal node (non-leaf node) has either one data element and two children or two data elements and three children. If a node contains one data element leftVal, it has two subtrees (children) namely left and middle. Whereas if a node contains two data elements leftVal and rightVal, it has three subtrees namely left, middle and right.

The main advantage with 2-3 trees is that it is balanced in nature as opposed to a binary search tree whose height in the worst case can be O(n). Due to this, the worst case time-complexity of operations such as search, insertion and deletion is as the height of a 2-3 tree is .

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:

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

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

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

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

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Improved By : DuyHoang