B-Trees

a B-tree is a tree data structure that keeps data sorted and allows searches, insertions, and deletions in logarithmic amortized time. It is most commonly used in databases and filesystems.

A B-tree is kept balanced by requiring that all leaf nodes are at the same depth. This depth will increase slowly as elements are added to the tree, but an increase in the overall depth is infrequent, and results in all leaf nodes being one more hop further removed from the root.

B-trees have substantial advantages over alternative implementations when node access times far exceed access times within nodes. This usually occurs when most nodes are in secondary storage such as hard drives. By maximizing the number of child nodes within each internal node, the height of the tree decreases, balancing occurs less often, and efficiency increases. Usually this value is set such that each node takes up a full disk block or an analogous size in secondary storage. While 2-3 B-trees might be useful in main memory, and are certainly easier to explain, if the node sizes are tuned to the size of a disk block, the result might be a 129-513 B-tree.

Search: Search is performed in the typical manner, analogous to that in a binary search tree. Starting at the root, the tree is traversed top to bottom, choosing the child pointer whose separation values are on either side of the value that is being searched. Binary search is typically (but not necessarily) used within nodes to find the separation values and child tree of interest.

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2-3 Tree

* Every non-leaf node has 2 or 3 children
* All data is kept in the leaves
* All leaves are at the same level (the bottom level)
* All data is kept in sorted order
* Every non-leaf node will contain 1 or 2 fields.

These contain one or two fields which indicate the range of values in its subtrees. If a node has two children, it will have one field; if the node has three children, it will have two fields. Each non-leaf node will contain a value in field 1 which is greater than the largest item in its left sub-tree, but less than or equal to the smallest item in its right sub-tree (or center sub-tree, if it has three children). If that node has three children, field 2 contains a value which is greater than the largest value in the center sub-tree, but less than or equal to the smallest item in its right sub-tree. The purpose of these values is to direct a search function to the correct sub-tree and eventually to the correct data node.

The biggest drawback with the 2-3 tree is that it requires more storage space than the normal binary search tree.
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2-3-4 Trees

A 2-3-4 tree is a B-tree of order 4.

2-3-4 trees are relatively difficult to implement in most programming languages because of the large number of special cases involved in operations on the tree. Red-black trees (see below) are simpler to implement, so they tend to be used instead.
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http://en.wikipedia.org/wiki/B-tree
http://en.wikipedia.org/wiki/2-3_tree