2–3 tree

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2-3 tree
Typetree
Invented1970
Invented byJohn Hopcroft
Time complexity in big O notation
Algorithm Average Worst case
Space O(n) O(n)
Search O(log n) O(log n)
Insert O(log n) O(log n)
Delete 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[edit]

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 non-empty 2–3 trees of the same height;
    • a is greater than each element in p; and
    • a is less than or equal to 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 non-empty 2–3 trees of equal height;
    • a is greater than each data element in p and less than or equal to each data element in q; and
    • b is greater than each data element in q and less than or equal to each data element in r.

Properties[edit]

  • 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[edit]

Searching[edit]

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 , then set T to p, which by definition is a 2–3 tree, and go back to step 2.
    • If , 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 . There are four cases:
    • If d is equal to a or b, then d is in T and we're done.
    • If , then set T to p and go back to step 2.
    • If , then set T to q and go back to step 2.
    • If , then set T to r and go back to step 2.

Insertion[edit]

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[edit]

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.

See also[edit]

References[edit]

  1. ^ Knuth, Donald M (1998). "6.2.4". The Art of Computer Programming. 3 (2 ed.). Addison Wesley. ISBN 9780201896855. The 2-3 trees defined at the close of Section 6.2.3 are equivalent to B-Trees of order 3.
  2. ^ Gross, R. Hernández, J. C. Lázaro, R. Dormido, S. Ros (2001). Estructura de Datos y Algoritmos. Prentice Hall. ISBN 84-205-2980-X.
  3. ^ Aho, Alfred V.; Hopcroft, John E.; Ullman, Jeffrey D. (1974). The Design and Analysis of Computer Algorithms. Addison-Wesley., p.145-147
  4. ^ Cormen, Thomas (2009). Introduction to Algorithms. London: The MIT Press. pp. 504. ISBN 978-0262033848.
  5. ^ a b Sedgewick, Robert; Wayne, Kevin. "3.3". Algorithms (4 ed.). Addison Wesley. ISBN 9780321573513.

External links[edit]