2–3 tree

2-3 tree
Type tree
Invented 1970
Invented by John 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. According to Knuth, "a B-tree of order 3 is a 2-3 tree." Nodes on the outside of the tree (leaf nodes) have no children and one or two data elements.[1][2] 2−3 trees were invented by John Hopcroft in 1970.[3]

2–3 trees are balanced, meaning that each right, center, and left subtree 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.

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

  • 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 L and right child R, then
    • L and R are non-empty 2–3 trees of the same height;
    • a is greater than each element in L; and
    • a is less than or equal to each data element in R.
  • T is a 3-node with data elements a and b, where a < b. If T has left child L, middle child M, and right child R, then
    • L, M, and R are non-empty 2–3 trees of equal height;
    • a is greater than each data element in L and less than or equal to each data element in M; and
    • b is greater than each data element in M and less than or equal to 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 r be the root of T.
  3. Suppose r is a leaf. If d is not in r, then d is not in T. Otherwise, d is in T. In particular, d can be found at a leaf node. We need no further steps and we're done.
  4. Suppose r is a 2-node with left child L and right child R. Let e be the data element in r. There are three cases:
    • If d is equal to e, then we've found d in T and we're done.
    • If , then set T to L, which by definition is a 2–3 tree, and go back to step 2.
    • If , then set T to R and go back to step 2.
  5. Suppose r is a 3-node with left child L, middle child M, and right child R. Let a and b be the two data elements of r, 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 L and go back to step 2.
    • If , then set T to M and go back to step 2.
    • If , then set T to R and go back to step 2.

Insertion

Insertion works by searching for the proper location of the key and adding it there. If the node becomes a 4-node then the node is split into two 2-nodes and the middle key is moved up to the parent. The diagram illustrates the process.

See also

References

  1. ↑ Gross, R. Hernández, J. C. Lázaro, R. Dormido, S. Ros (2001). Estructura de Datos y Algoritmos. Prentice Hall. ISBN 84-205-2980-X.
  2. ↑ Aho, Alfred V.; Hopcroft, John E.; Ullman, Jeffrey D. (1974). The Design and Analysis of Computer Algorithms. Addison-Wesley. , p.145-147
  3. ↑ Cormen, Thomas (2009). Introduction to Algorithms. London: The MIT Press. p. 504. ISBN 978-0262033848.
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