B+ tree

A B+ tree is an n-ary tree with a variable but often large number of children per node. A B+ tree consists of a root, internal nodes and leaves. The root may be either a leaf or a node with two or more children.

A B+ tree can be viewed as a B-tree in which each node contains only keys (not key-value pairs), and to which an additional level is added at the bottom with linked leaves.

The primary value of a B+ tree is in storing data for efficient retrieval in a block-oriented storage context — in particular, filesystems. This is primarily because unlike binary search trees, B+ trees have very high fanout (number of pointers to child nodes in a node, typically on the order of 100 or more), which reduces the number of I/O operations required to find an element in the tree.

The ReiserFS, NSS, XFS, JFS, ReFS, and BFS filesystems all use this type of tree for metadata indexing; BFS also uses B+ trees for storing directories. NTFS uses B+ trees for directory indexing. EXT4 uses extent trees (a modified B+ tree data structure) for file extent indexing. Relational database management systems such as IBM DB2,Informix,Microsoft SQL Server,Oracle 8,Sybase ASE, and SQLite support this type of tree for table indices. Key-value database management systems such as CouchDB and Tokyo Cabinet support this type of tree for data access.

B-tree

In computer science, a B-tree is a self-balancing tree data structure that keeps data sorted and allows searches, sequential access, insertions, and deletions in logarithmic time. The B-tree is a generalization of a binary search tree in that a node can have more than two children (Comer 1979, p. 123). Unlike self-balancing binary search trees, the B-tree is optimized for systems that read and write large blocks of data. B-trees are a good example of a data structure for external memory. It is commonly used in databases and filesystems.

Overview

In B-trees, internal (non-leaf) nodes can have a variable number of child nodes within some pre-defined range. When data is inserted or removed from a node, its number of child nodes changes. In order to maintain the pre-defined range, internal nodes may be joined or split. Because a range of child nodes is permitted, B-trees do not need re-balancing as frequently as other self-balancing search trees, but may waste some space, since nodes are not entirely full. The lower and upper bounds on the number of child nodes are typically fixed for a particular implementation. For example, in a 2-3 B-tree (often simply referred to as a 2-3 tree), each internal node may have only 2 or 3 child nodes.

Ṭūbā

Ṭūbā (Arabic: طُوبَىٰ‎ ṭūbā, lit. "blessedness") is a tree that Muslims believe grows in Jannah, or Islamic heaven.

The tree is only mentioned once in the Koran but its attributes can be ascertained from various hadiths and other writings. Ṣaḥīḥ al-Bukhārī 4:474 notes that the tree spreads over a distance that can be covered in a hundred years and that the gowns of righteous dwellers of heaven are made from the fiber of its blossoms. A sage in Sohrevardi's The Red Intellect says that "any fruit you see in the world will be on that tree". The tree is sometimes described as being inverted and having its roots in the air. Mehmed Yazıcıoğlu (Yazıcızade, Yaziji-Oglu) wrote of the tree in "The Creation of Paradise" in his 1449 Book of Muhammad (Muhammediye):

That a tree which hangeth downward, high aloft its roots are there: Thus its radiance all the Heavens lighteth up from end to end, Flooding every tent and palace, every lane and every square. Such a tree the Tuba, that the Gracious One hath in its sap Hidden whatsoe'er there be of gifts and presents good and fair; Forth therefrom crowns, thrones, and jewels, yea, and steeds and coursers come, Golden leaves and clearest crystals, wines most pure beyond compare. For his sake there into being hath he called the Tuba-Tree, That from Ebu-Qasim's hand might every one receive his share.