B-tree

The literature on B-trees is not uniform in its terminology.[9]

Bayer and McCreight (1972),[3] Comer (1979),[2] and others define the order of B-tree as the minimum number of keys in a non-root node. Folk and Zoellick [10] points out that terminology is ambiguous because the maximum number of keys is not clear. An order 3 B-tree might hold a maximum of 6 keys or a maximum of 7 keys. Knuth (1998) avoids the problem by defining the order to be the maximum number of children (which is one more than the maximum number of keys).[8]

The term leaf is also inconsistent. Bayer and McCreight (1972)[3] considered the leaf level to be the lowest level of keys, but Knuth considered the leaf level to be one level below the lowest keys.[11] There are many possible implementation choices. In some designs, the leaves may hold the entire data record; in other designs, the leaves may only hold pointers to the data record. Those choices are not fundamental to the idea of a B-tree.[12]

For simplicity, most authors assume there are a fixed number of keys that fit in a node. The basic assumption is the key size is fixed and the node size is fixed. In practice, variable length keys may be employed.[13]

As with other trees, B-trees can be represented as a collection of three types of nodes: root, internal (a.k.a. interior), and leaf.

Note the following variable definitions:

• K: Maximum number of potential search keys for each node in a B-tree. (this value is constant over the entire tree).
• ${\displaystyle pt_{i}}$: The pointer to a child node which starts a sub-tree.
• ${\displaystyle pr_{i}}$: The pointer to a record which stores the data.
• ${\displaystyle k_{i}}$: The search key at the zero-based node index ${\displaystyle i}$.

In B-trees, the following properties are maintained for these nodes:

• If ${\textstyle k_{i}}$ exists in any node in a B+ tree, then ${\displaystyle k_{i-1}}$ exists in that node where ${\displaystyle i\geq 1}$.
• All leaf nodes have the same number of ancestors (i.e., they are all at the same depth).

Each internal node in a B-tree has the following format:

Internal Node Pointer Structure

${\displaystyle pt_{0}}$ when ${\displaystyle k_{0}}$ exists	${\displaystyle pt_{i}}$ when ${\displaystyle k_{i-1}}$ and ${\displaystyle k_{i}}$ exist	${\displaystyle pt_{i}}$ when ${\displaystyle k_{i-1}}$ exists, and ${\displaystyle k_{i}}$ does not exist	${\displaystyle pt_{i}}$ when ${\displaystyle k_{i-1}}$ and ${\displaystyle k_{i}}$ do not exist	${\displaystyle pr_{i}}$ when ${\displaystyle k_{i}}$ exists	${\displaystyle pr_{i}}$ when ${\displaystyle k_{i}}$ does not exist
Points to subtree in which all search keys are less than ${\displaystyle k_{0}}$.	Points to subtree in which all search keys are greater than ${\displaystyle k_{i-1}}$ and are less than ${\displaystyle k_{i}}$.	Points to subtree in which all search keys are greater than ${\displaystyle k_{i-1}}$.	Here, ${\displaystyle pt_{i}}$ is empty.	Points to a record with a value equal to ${\displaystyle k_{i}}$.	Here, ${\displaystyle pr_{i}}$ is empty.

Each leaf node in a B-tree has the following format:

Leaf Node Pointer Structure

${\displaystyle pr_{i}}$ when ${\displaystyle k_{i}}$ exists	${\displaystyle pr_{i}}$ when ${\displaystyle k_{i}}$ does not exist
Points to a record with a value equal to ${\displaystyle k_{i}}$.	Here, ${\displaystyle pr_{i}}$ is empty.

The node bounds are summarized in the table below:

Node Type	Min Number of Keys	Max Number of Keys	Min Number of Child Nodes	Max Number of Child Nodes
Root Node (when it is a leaf node)	0	${\displaystyle K}$	0	0
Root Node (when it is an internal node)	1	${\displaystyle K}$	2[14]	${\displaystyle K+1}$
Internal Node	${\displaystyle \lfloor K/2\rfloor }$	${\displaystyle K}$	${\displaystyle \lceil (K+1)/2\rceil }$ ${\displaystyle \equiv \lfloor K/2\rfloor +1}$	${\displaystyle K+1}$
Leaf Node	${\displaystyle \lceil K/2\rceil }$	${\displaystyle K}$	0	0

In order to maintain the pre-defined range of child nodes, internal nodes may be joined or split.

Usually, the number of keys is chosen to vary between ${\displaystyle d}$ and ${\displaystyle 2d}$, where ${\displaystyle d}$ is the minimum number of keys, and ${\displaystyle d+1}$ is the minimum degree or branching factor of the tree. The factor of 2 will guarantee that nodes can be split or combined.

If an internal node has ${\displaystyle 2d}$ keys, then adding a key to that node can be accomplished by splitting the hypothetical ${\displaystyle 2d+1}$ key node into two ${\displaystyle d}$ key nodes and moving the key that would have been in the middle to the parent node. Each split node has the required minimum number of keys. Similarly, if an internal node and its neighbor each have ${\displaystyle d}$ keys, then a key may be deleted from the internal node by combining it with its neighbor. Deleting the key would make the internal node have ${\displaystyle d-1}$ keys; joining the neighbor would add ${\displaystyle d}$ keys plus one more key brought down from the neighbor's parent. The result is an entirely full node of ${\displaystyle 2d}$ keys.

A B-tree is kept balanced after insertion by splitting a would-be overfilled node, of ${\displaystyle 2d+1}$ keys, into two ${\displaystyle d}$-key siblings and inserting the mid-value key into the parent. Depth only increases when the root is split, maintaining balance. Similarly, a B-tree is kept balanced after deletion by merging or redistributing keys among siblings to maintain the ${\displaystyle d}$-key minimum for non-root nodes. A merger reduces the number of keys in the parent potentially forcing it to merge or redistribute keys with its siblings, and so on. The only change in depth occurs when the root has two children, of ${\displaystyle d}$ and (transitionally) ${\displaystyle d-1}$ keys, in which case the two siblings and parent are merged, reducing the depth by one.

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 node farther away from the root.

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.

B-trees have substantial advantages over alternative implementations when the time to access the data of a node greatly exceeds the time spent processing that data, because then the cost of accessing the node may be amortized over multiple operations within the node. This usually occurs when the node data are in secondary storage such as disk drives. By maximizing the number of keys within each internal node, the height of the tree decreases and the number of expensive node accesses is reduced. In addition, rebalancing of the tree occurs less often. The maximum number of child nodes depends on the information that must be stored for each child node and the size of a full disk block or an analogous size in secondary storage. While 2–3 B-trees are easier to explain, practical B-trees using secondary storage need a large number of child nodes to improve performance.

The term B-tree may refer to a specific design or it may refer to a general class of designs. In the narrow sense, a B-tree stores keys in its internal nodes but need not store those keys in the records at the leaves. The general class includes variations such as the B+ tree, the B^* tree and the B^*+ tree.

• In the B+ tree, the internal nodes do not store any pointers to records, thus all pointers to records are stored in the leaf nodes. In addition, a leaf node may include a pointer to the next leaf node to speed sequential access.[2] Because B+ tree internal nodes have fewer pointers, each node can hold more keys, causing the tree to be shallower, and thus, faster to search.
• The B^* tree balances more neighboring internal nodes to keep the internal nodes more densely packed.[2] This variant ensures non-root nodes are at least 2/3 full instead of 1/2.[15] As the most costly part of operation of inserting the node in B-tree is splitting the node, B^*-trees are created to postpone splitting operation as long as they can.[16] To maintain this, instead of immediately splitting up a node when it gets full, its keys are shared with a node next to it. This spill operation is less costly to do than split, because it requires only shifting the keys between existing nodes, not allocating memory for a new one.[16] For inserting, first it is checked whether the node has some free space in it, and if so, the new key is just inserted in the node. However, if the node is full (it has m − 1 keys, where m is the order of the tree as maximum number of pointers to subtrees from one node), it needs to be checked whether the right sibling exists and has some free space. If the right sibling has j < m − 1 keys, then keys are redistributed between the two sibling nodes as evenly as possible. For this purpose, m - 1 keys from the current node, the new key inserted, one key from the parent node and j keys from the sibling node are seen as an ordered array of m + j + 1 keys. The array becomes split by half, so that ⌊(m + j + 1)/2⌋ lowest keys stay in the current node, the next (middle) key is inserted in the parent and the rest go to the right sibling.[16] (The newly inserted key might end up in any of the three places.) The situation when right sibling is full, and left isn't is analogous.[16] When both the sibling nodes are full, then the two nodes (current node and a sibling) are split into three and one more key is shifted up the tree, to the parent node.[16] If the parent is full, then spill/split operation propagates towards the root node.[16] Deleting nodes is somewhat more complex than inserting however.
• The B^*+ tree combines the main B+ tree and B^* tree features together.[17]
• B-trees can be turned into order statistic trees to allow rapid searches for the Nth record in key order, or counting the number of records between any two records, and various other related operations.[18]

Usually, sorting and searching algorithms have been characterized by the number of comparison operations that must be performed using order notation. A binary search of a sorted table with N records, for example, can be done in roughly ⌈ log₂ N ⌉ comparisons. If the table had 1,000,000 records, then a specific record could be located with at most 20 comparisons: ⌈ log₂ (1,000,000) ⌉ = 20.

Large databases have historically been kept on disk drives. The time to read a record on a disk drive far exceeds the time needed to compare keys once the record is available. The time to read a record from a disk drive involves a seek time and a rotational delay. The seek time may be 0 to 20 or more milliseconds, and the rotational delay averages about half the rotation period. For a 7200 RPM drive, the rotation period is 8.33 milliseconds. For a drive such as the Seagate ST3500320NS, the track-to-track seek time is 0.8 milliseconds and the average reading seek time is 8.5 milliseconds.[19] For simplicity, assume reading from disk takes about 10 milliseconds.

Naively, then, the time to locate one record out of a million would take 20 disk reads times 10 milliseconds per disk read, which is 0.2 seconds.

The time won't be that bad because individual records are grouped together in a disk block. A disk block might be 16 kilobytes. If each record is 160 bytes, then 100 records could be stored in each block. The disk read time above was actually for an entire block. Once the disk head is in position, one or more disk blocks can be read with little delay. With 100 records per block, the last 6 or so comparisons don't need to do any disk reads—the comparisons are all within the last disk block read.

To speed the search further, the first 13 to 14 comparisons (which each required a disk access) must be sped up.

A significant improvement in performance can be made with a B-tree index. A B-tree index creates a multi-level tree structure that breaks a database down into fixed-size blocks or pages. Each level of this tree can be used to link those pages via an address location, allowing one page (known as a node, or internal page) to refer to another with leaf pages at the lowest level. One page is typically the starting point of the tree, or the "root". This is where the search for a particular key would begin, traversing a path that terminates in a leaf. Most pages in this structure will be leaf pages which ultimately refer to specific table rows.

Because each node (or internal page) can have more than two children, a B-tree index will usually have a shorter height (the distance from the root to the farthest leaf) than a Binary Search Tree. In the example above, initial disk reads narrowed the search range by a factor of two. That can be improved substantially by creating an auxiliary index that contains the first record in each disk block (sometimes called a sparse index). This auxiliary index would be 1% of the size of the original database, but it can be searched more quickly. Finding an entry in the auxiliary index would tell us which block to search in the main database; after searching the auxiliary index, we would have to search only that one block of the main database—at a cost of one more disk read. The index would hold 10,000 entries, so it would take at most 14 comparisons. Like the main database, the last six or so comparisons in the auxiliary index would be on the same disk block. The index could be searched in about eight disk reads, and the desired record could be accessed in 9 disk reads.

The trick of creating an auxiliary index can be repeated to make an auxiliary index to the auxiliary index. That would make an aux-aux index that would need only 100 entries and would fit in one disk block.

Instead of reading 14 disk blocks to find the desired record, we only need to read 3 blocks. This blocking is the core idea behind the creation of the B-tree, where the disk blocks fill-out a hierarchy of levels to make up the index. Reading and searching the first (and only) block of the aux-aux index which is the root of the tree identifies the relevant block in aux-index in the level below. Reading and searching that aux-index block identifies the relevant block to read, until the final level, known as the leaf level, identifies a record in the main database. Instead of 150 milliseconds, we need only 30 milliseconds to get the record.

The auxiliary indices have turned the search problem from a binary search requiring roughly log₂ N disk reads to one requiring only log_b N disk reads where b is the blocking factor (the number of entries per block: b = 100 entries per block in our example; log₁₀₀ 1,000,000 = 3 reads).

In practice, if the main database is being frequently searched, the aux-aux index and much of the aux index may reside in a disk cache, so they would not incur a disk read. The B-tree remains the standard index implementation in almost all relational databases, and many nonrelational databases use them too.[20]

If the database does not change, then compiling the index is simple to do, and the index need never be changed. If there are changes, then managing the database and its index becomes more complicated.

Deleting records from a database is relatively easy. The index can stay the same, and the record can just be marked as deleted. The database remains in sorted order. If there are a large number of lazy deletions, then searching and storage become less efficient.[21]

Insertions can be very slow in a sorted sequential file because room for the inserted record must be made. Inserting a record before the first record requires shifting all of the records down one. Such an operation is just too expensive to be practical. One solution is to leave some spaces. Instead of densely packing all the records in a block, the block can have some free space to allow for subsequent insertions. Those spaces would be marked as if they were "deleted" records.

Both insertions and deletions are fast as long as space is available on a block. If an insertion won't fit on the block, then some free space on some nearby block must be found and the auxiliary indices adjusted. The hope is that enough space is available nearby, such that a lot of blocks do not need to be reorganized. Alternatively, some out-of-sequence disk blocks may be used.[20]

The B-tree uses all of the ideas described above. In particular, a B-tree:

• keeps keys in sorted order for sequential traversing
• uses a hierarchical index to minimize the number of disk reads
• uses partially full blocks to speed up insertions and deletions
• keeps the index balanced with a recursive algorithm

In addition, a B-tree minimizes waste by making sure the interior nodes are at least half full. A B-tree can handle an arbitrary number of insertions and deletions.[20]

Searching is similar to searching a binary search tree. Starting at the root, the tree is recursively traversed from top to bottom. At each level, the search reduces its field of view to the child pointer (subtree) whose range includes the search value. A subtree's range is defined by the values, or keys, contained in its parent node. These limiting values are also known as separation values.

Binary search is typically (but not necessarily) used within nodes to find the separation values and child tree of interest.

A B-tree insertion example with each iteration. The nodes of this B-tree have at most 3 children (Knuth order 3).

All insertions start at a leaf node. To insert a new element, search the tree to find the leaf node where the new element should be added. Insert the new element into that node with the following steps:

1. If the node contains fewer than the maximum allowed number of elements, then there is room for the new element. Insert the new element in the node, keeping the node's elements ordered.
2. Otherwise the node is full, evenly split it into two nodes so:
   1. A single median is chosen from among the leaf's elements and the new element that is being inserted.
   2. Values less than the median are put in the new left node and values greater than the median are put in the new right node, with the median acting as a separation value.
   3. The separation value is inserted in the node's parent, which may cause it to be split, and so on. If the node has no parent (i.e., the node was the root), create a new root above this node (increasing the height of the tree).

If the splitting goes all the way up to the root, it creates a new root with a single separator value and two children, which is why the lower bound on the size of internal nodes does not apply to the root. The maximum number of elements per node is U−1. When a node is split, one element moves to the parent, but one element is added. So, it must be possible to divide the maximum number U−1 of elements into two legal nodes. If this number is odd, then U=2L and one of the new nodes contains (U−2)/2 = L−1 elements, and hence is a legal node, and the other contains one more element, and hence it is legal too. If U−1 is even, then U=2L−1, so there are 2L−2 elements in the node. Half of this number is L−1, which is the minimum number of elements allowed per node.

An alternative algorithm supports a single pass down the tree from the root to the node where the insertion will take place, splitting any full nodes encountered on the way pre-emptively. This prevents the need to recall the parent nodes into memory, which may be expensive if the nodes are on secondary storage. However, to use this algorithm, we must be able to send one element to the parent and split the remaining U−2 elements into two legal nodes, without adding a new element. This requires U = 2L rather than U = 2L−1, which accounts for why some textbooks impose this requirement in defining B-trees.

There are two popular strategies for deletion from a B-tree.

1. Locate and delete the item, then restructure the tree to retain its invariants, OR
2. Do a single pass down the tree, but before entering (visiting) a node, restructure the tree so that once the key to be deleted is encountered, it can be deleted without triggering the need for any further restructuring

The algorithm below uses the former strategy.

There are two special cases to consider when deleting an element:

1. The element in an internal node is a separator for its child nodes
2. Deleting an element may put its node under the minimum number of elements and children

The procedures for these cases are in order below.

1. Search for the value to delete.
2. If the value is in a leaf node, simply delete it from the node.
3. If underflow happens, rebalance the tree as described in section "Rebalancing after deletion" below.

Each element in an internal node acts as a separation value for two subtrees, therefore we need to find a replacement for separation. Note that the largest element in the left subtree is still less than the separator. Likewise, the smallest element in the right subtree is still greater than the separator. Both of those elements are in leaf nodes, and either one can be the new separator for the two subtrees. Algorithmically described below:

1. Choose a new separator (either the largest element in the left subtree or the smallest element in the right subtree), remove it from the leaf node it is in, and replace the element to be deleted with the new separator.
2. The previous step deleted an element (the new separator) from a leaf node. If that leaf node is now deficient (has fewer than the required number of nodes), then rebalance the tree starting from the leaf node.

Rebalancing starts from a leaf and proceeds toward the root until the tree is balanced. If deleting an element from a node has brought it under the minimum size, then some elements must be redistributed to bring all nodes up to the minimum. Usually, the redistribution involves moving an element from a sibling node that has more than the minimum number of nodes. That redistribution operation is called a rotation. If no sibling can spare an element, then the deficient node must be merged with a sibling. The merge causes the parent to lose a separator element, so the parent may become deficient and need rebalancing. The merging and rebalancing may continue all the way to the root. Since the minimum element count doesn't apply to the root, making the root be the only deficient node is not a problem. The algorithm to rebalance the tree is as follows:

• If the deficient node's right sibling exists and has more than the minimum number of elements, then rotate left
  1. Copy the separator from the parent to the end of the deficient node (the separator moves down; the deficient node now has the minimum number of elements)
  2. Replace the separator in the parent with the first element of the right sibling (right sibling loses one node but still has at least the minimum number of elements)
  3. The tree is now balanced
• Otherwise, if the deficient node's left sibling exists and has more than the minimum number of elements, then rotate right
  1. Copy the separator from the parent to the start of the deficient node (the separator moves down; deficient node now has the minimum number of elements)
  2. Replace the separator in the parent with the last element of the left sibling (left sibling loses one node but still has at least the minimum number of elements)
  3. The tree is now balanced
• Otherwise, if both immediate siblings have only the minimum number of elements, then merge with a sibling sandwiching their separator taken off from their parent
  1. Copy the separator to the end of the left node (the left node may be the deficient node or it may be the sibling with the minimum number of elements)
  2. Move all elements from the right node to the left node (the left node now has the maximum number of elements, and the right node – empty)
  3. Remove the separator from the parent along with its empty right child (the parent loses an element)
     • If the parent is the root and now has no elements, then free it and make the merged node the new root (tree becomes shallower)
     • Otherwise, if the parent has fewer than the required number of elements, then rebalance the parent[23]

Note: The rebalancing operations are different for B+ trees (e.g., rotation is different because parent has copy of the key) and B^*-tree (e.g., three siblings are merged into two siblings).

While freshly loaded databases tend to have good sequential behaviour, this behaviour becomes increasingly difficult to maintain as a database grows, resulting in more random I/O and performance challenges.[24]

A common special case is adding a large amount of pre-sorted data into an initially empty B-tree. While it is quite possible to simply perform a series of successive inserts, inserting sorted data results in a tree composed almost entirely of half-full nodes. Instead, a special "bulk loading" algorithm can be used to produce a more efficient tree with a higher branching factor.

When the input is sorted, all insertions are at the rightmost edge of the tree, and in particular any time a node is split, we are guaranteed that no more insertions will take place in the left half. When bulk loading, we take advantage of this, and instead of splitting overfull nodes evenly, split them as unevenly as possible: leave the left node completely full and create a right node with zero keys and one child (in violation of the usual B-tree rules).

At the end of bulk loading, the tree is composed almost entirely of completely full nodes; only the rightmost node on each level may be less than full. Because those nodes may also be less than half full, to re-establish the normal B-tree rules, combine such nodes with their (guaranteed full) left siblings and divide the keys to produce two nodes at least half full. The only node which lacks a full left sibling is the root, which is permitted to be less than half full.

Lehman and Yao[27] showed that all the read locks could be avoided (and thus concurrent access greatly improved) by linking the tree blocks at each level together with a "next" pointer. This results in a tree structure where both insertion and search operations descend from the root to the leaf. Write locks are only required as a tree block is modified. This maximizes access concurrency by multiple users, an important consideration for databases and/or other B-tree-based ISAM storage methods. The cost associated with this improvement is that empty pages cannot be removed from the btree during normal operations. (However, see [28] for various strategies to implement node merging, and source code at.[29])

United States Patent 5283894, granted in 1994, appears to show a way to use a 'Meta Access Method' [30] to allow concurrent B+ tree access and modification without locks. The technique accesses the tree 'upwards' for both searches and updates by means of additional in-memory indexes that point at the blocks in each level in the block cache. No reorganization for deletes is needed and there are no 'next' pointers in each block as in Lehman and Yao.

Since B-trees are similar in structure to red-black trees, parallel algorithms for red-black trees can be applied to B-trees as well.

A Maple tree is a B-tree developed for use in the Linux kernel to reduce lock contention in virtual memory management.[31][32][33]

• Bayer, Rudolf; McCreight, E. (July 1970), Organization and Maintenance of Large Ordered Indices, vol. Mathematical and Information Sciences Report No. 20, Boeing Scientific Research Laboratories.
• Bayer, Rudolf (1971), Binary B-Trees for Virtual Memory, Proceedings of 1971 ACM-SIGFIDET Workshop on Data Description, Access and Control, San Diego, California{{citation}}: CS1 maint: location missing publisher (link).