# 2–3 tree

2–3 tree
Typetree
Invented1970
Invented byJohn Hopcroft
Time complexity in big O notation
Algorithm Space Average Worst case O(n) O(n) O(log n) O(log n) O(log n) O(log n) O(log n) O(log n)

In computer science, a 2–3 tree is a tree data structure, where every node with children (internal node) has either two children (2-node) and one data element or three children (3-nodes) and two data elements. A 2–3 tree is a B-tree of order 3.[1] Nodes on the outside of the tree (leaf nodes) have no children and one or two data elements.[2][3] 2–3 trees were invented by John Hopcroft in 1970.[4]

2–3 trees are required to be balanced, meaning that each leaf is at the same level. It follows that each right, center, and left subtree of a node contains the same or close to the same amount of data.

## Definitions

We say that an internal node is a 2-node if it has one data element and two children.

We say that an internal node is a 3-node if it has two data elements and three children.

A 4-node, with three data elements, may be temporarily created during manipulation of the tree but is never persistently stored in the tree.

We say that T is a 2–3 tree if and only if one of the following statements hold:[5]

• T is empty. In other words, T does not have any nodes.
• T is a 2-node with data element a. If T has left child p and right child q, then
• p and q are 2–3 trees of the same height;
• a is greater than each element in p; and
• a is less than each data element in q.
• T is a 3-node with data elements a and b, where a < b. If T has left child p, middle child q, and right child r, then
• p, q, and r are 2–3 trees of equal height;
• a is greater than each data element in p and less than each data element in q; and
• b is greater than each data element in q and less than each data element in r.

## Properties

• Every internal node is a 2-node or a 3-node.
• All leaves are at the same level.
• All data is kept in sorted order.

## Operations

### Searching

Searching for an item in a 2–3 tree is similar to searching for an item in a binary search tree. Since the data elements in each node are ordered, a search function will be directed to the correct subtree and eventually to the correct node which contains the item.

1. Let T be a 2–3 tree and d be the data element we want to find. If T is empty, then d is not in T and we're done.
2. Let t be the root of T.
3. Suppose t is a leaf.
• If d is not in t, then d is not in T. Otherwise, d is in T. We need no further steps and we're done.
4. Suppose t is a 2-node with left child p and right child q. Let a be the data element in t. There are three cases:
• If d is equal to a, then we've found d in T and we're done.
• If ${\displaystyle d, then set T to p, which by definition is a 2–3 tree, and go back to step 2.
• If ${\displaystyle d>a}$, then set T to q and go back to step 2.
5. Suppose t is a 3-node with left child p, middle child q, and right child r. Let a and b be the two data elements of t, where ${\displaystyle a. There are four cases:
• If d is equal to a or b, then d is in T and we're done.
• If ${\displaystyle d, then set T to p and go back to step 2.
• If ${\displaystyle a, then set T to q and go back to step 2.
• If ${\displaystyle d>b}$, then set T to r and go back to step 2.

### Insertion

Insertion maintains the balanced property of the tree.[5]

To insert into a 2-node, the new key is added to the 2-node in the appropriate order.

To insert into a 3-node, more work may be required depending on the location of the 3-node. If the tree consists only of a 3-node, the node is split into three 2-nodes with the appropriate keys and children.

Insertion of a number in a 2–3 tree for 3 possible cases

If the target node is a 3-node whose parent is a 2-node, the key is inserted into the 3-node to create a temporary 4-node. In the illustration, the key 10 is inserted into the 2-node with 6 and 9. The middle key is 9, and is promoted to the parent 2-node. This leaves a 3-node of 6 and 10, which is split to be two 2-nodes held as children of the parent 3-node.

If the target node is a 3-node and the parent is a 3-node, a temporary 4-node is created then split as above. This process continues up the tree to the root. If the root must be split, then the process of a single 3-node is followed: a temporary 4-node root is split into three 2-nodes, one of which is considered to be the root. This operation grows the height of the tree by one.

### Parallel operations

Since 2–3 trees are similar in structure to red–black trees, parallel algorithms for red–black trees can be applied to 2–3 trees as well.