2-3 Insertion

Introduction

Insertion in 2-3 Trees is a lot different from the insertion in Binary Search Trees. In 2-3 Trees, values are only inserted at leaf nodes at certain conditions. As discussed before, the insertion algorithm takes $O(log\ n)$ time where n is the number of nodes in the tree. Searching an element is done in $log(n)$ and then insertion takes a constant amount of time. So overall the time complexity of the insertion algorithm is $O(log\ n)$. Let’s see how it works.

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Insertion Algorithm

The insertion algorithm is based on these scenarios:

• Initially if the tree is empty, create a new leaf node and insert your value
• If the tree is not empty, traverse through the tree to find the right leaf node where the value should be inserted
• If the leaf node has only one value, insert your value into the node
• If the leaf node has more than two values, split the node by moving the middle element to the top node
• Keep forming new nodes wherever you get more than two elements

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Example 1

Let’s take a look at the following example where we will build a 2-3 Tree from scratch by inserting elements one by one.