Tree

Tree

[tree]
Tree, Ellen: see Kean, Edmund.
Tree, Sir Herbert Beerbohm, 1853-1917, English actor-manager, whose original name was Herbert Draper Beerbohm. He was a half brother of Max Beerbohm. His first success (1884) was as the curate in The Private Secretary, and he thereafter became prominent as a romantic actor. In 1883 he married the distinguished actress Helen Maud Holt, 1863-1937, who became his leading lady. She was a well-educated and very versatile actress, especially adept at comedy. Tree achieved his greatest distinction as a manager with his staging of Shakespeare at the Haymarket theater (1887-97) and at Her Majesty's Theatre, which he built and opened in 1897. In the manner of his day, he stressed visual elements with elaborate, imaginative, and detailed effects. He was knighted in 1909.

See his Thoughts and Afterthoughts (1913) and Nothing Matters (1917); biography by H. Pearson (1956, repr. 1971).

tree, perennial woody plant with a single main stem (the trunk, or bole) from which branches and twigs extend to form a characteristic crown of foliage. In general, a tree differs from a shrub in that it has a single trunk, it reaches a greater height at maturity, it branches at a greater distance from the ground, and it increases in size by producing new branches and expanding in girth while a shrub often produces new shoots from ground level. Trees fall into three major divisions: angiosperms, gymnosperms, and pteridophytes. Angiosperms are the most common type, where seeds carried in various fruits are the agents of reproduction. Trees and shrubs may be deciduous, with broad leaves that are shed at the end of the growing season, or evergreen (see conifer), with needlelike or scalelike leaves that are shed at intervals of between 2 and 10 years, thus maintaining green foliage at all seasons. Trees are identified both by the characteristic color and shape of the leaf and by their overall appearance, e.g., the degree and angle of branching, the shape of the crown, and the texture of the bark. Their age can be determined from a count of the annual rings, which represent the diameter growth of a tree each year. Besides their enormous importance in providing oxygen and moisture for the atmosphere and removing harmful carbon dioxide, trees are an important source of food, of wood, and of numerous products (e.g., resins, rubber, quinine, turpentine, and cellulose for the manufacture of paper and various synthetic materials) derived from their wood, bark, leaves, and fruits.

See H. Johnson, The International Book of Trees (1973); L. Line and A. Sutton, Audubon Society Book of Trees (1981); A. C. Barefoot and F. W. Hankins, Identification of Modern Tertiary Woods (1982).

Centre of the world, a widespread motif in myths and folktales among various peoples, especially in Asia, Australia, and North America. There are two main forms. In the vertical tradition, the tree extends between and connects earth, heaven, and the underworld; oracles, judgments, and other prophetic activities are performed at its base. In the horizontal tradition, the tree is planted at the centre of the world and is protected by supernatural guardians; it is the source of terrestrial fertility and life.

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or tulip poplar or yellow poplar

Lofty North American ornamental and timber tree (Liriodendron tulipifera) of the magnolia family, not related to true poplars. It occurs in mixed hardwood stands in eastern North America. It is taller than all other eastern broad-leaved trees (up to 197 ft, or 60 m), and its trunk often has a diameter greater than 7 ft (2 m). Long-stemmed, bright-green leaves have two to four side lobes and blunted tips. Yellowish-green tuliplike flowers have six petals, orange at their bases, and three bright-green sepals. Other characteristics include conelike clusters of winged fruits; aromatic, purplish-brown twigs; stunning golden-yellow autumn leaves; winter buds resembling a duck's bill; and resistance to pests and diseases. The wood is used to manufacture furniture parts, plywood panels, paper, boxes, and crates.

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or tree toad

European green tree frogs (Hyla arborea).

Any of some 550 species (family Hylidae) of mostly arboreal frogs, found worldwide but primarily in the New World. Most species are small, slender, and long-legged and have suckerlike adhesive disks on the finger and toe tips. Some do not climb well and live in water, on land, or in a burrow. Most species lay eggs in water. Young marsupial frogs (genus Gastrotheca), of South America, develop in a brood pouch on the female's back.

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Three-toed sloth (Bradypus tridactylus)

Nocturnal, solitary, tree-dwelling mammal (family Bradypodidae), found in South and Central America. About 2 ft (60 cm) long, sloths have a tiny tail, peglike teeth, long curved claws, and long forelimbs. A green alga grows in the shaggy fur. The four species of three-toed sloths, or ais (Bradypus), eat only leaves of the trumpet tree. The two species of two-toed sloths, or unaus (Choloepus), have two toes on the forelimbs; they eat fruits, stems, and leaves of various plants. Sloths cannot walk. They cling upright to trunks, hang upside down (in which position they sleep some 15 hours a day), or move, extremely slowly (hence their name), by pulling hand over hand. Their natural camouflage is their chief protection from predators.

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(Latin: “tree of life”) Any of six species (genus Thuja) of resinous, evergreen ornamental and timber conifers of the cypress family, native to North America and eastern Asia. Arborvitae trees or shrubs have thin, scaling outer bark and fibrous inner bark; horizontal or ascending branches; and flattened, spraylike branchlets with scalelike leaves. The Oriental, or Chinese, arborvitae (T. orientalis), a popular ornamental native to Asia, is a gracefully symmetrical shrub. Arborvitae wood is soft and lightweight but very durable, fragrant, and easily worked. The giant arborvitae (T. plicata) is the most important timber-producing species, but the wood of the American arborvitae (T. occidentalis) is also frequently used. False arborvitae (Thujopsis dolabrata) is closely related.

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Rapid-growing tree (Ailanthus altissima) in the quassia family (Simaroubaceae), native to China and widely naturalized elsewhere, with several known varieties. Because of its resistance to pollution, freedom from insect predation and disease, and ability to grow in almost any soil, the tree of heaven is planted as a yard and street tree in urban centres. It grows to 60 ft (18 m) or more, producing long, compound leaves that emit an unpleasant odour when bruised. Male trees bear flowers with unpleasant scents. Female trees produce winged fruits which are tannish orange when ripe.

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Woody perennial plant. Most trees have a single self-supporting trunk containing woody tissues, and in most species the trunk produces secondary limbs called branches. Trees provide many valuable products, especially wood, one of the world's chief building materials, and wood pulp, used in papermaking. Wood is also a major fuel source. Trees supply edible fruits and nuts. In addition, trees take in carbon dioxide and release oxygen during photosynthesis. Their root systems help retain water and soil, preventing floods and erosion. Trees and forests provide habitats for a wide variety of animals, and they beautify both natural and altered landscapes. Growth rings in the trunk indicate the age of most trees. The tallest trees are the Pacific coast redwoods; the oldest are the bristlecone pines, some of which are over 4,000 years old. Seealso conifer; deciduous tree; evergreen; forest; shrub; softwood.

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or India rubber plant

Tropical tree (Ficus elastica) of the mulberry family. The rubber plant is large in its native Southeast Asia and other warm areas; elsewhere it is commonly grown indoors as a potted plant. The plant has large, thick, oblong leaves and pairs of figlike fruits along its branches. The milky sap, or latex, was once an important source of an inferior natural rubber. Young plants available in the florist's trade are durable and grow well under less-than-ideal indoor conditions. Some cultivated varieties have broader, darker green leaves; others are variegated. Seealso rubber tree.

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Any of ten species of large trees that make up the genus Platanus, sole genus of the family Platanaceae, native to North America, eastern Europe, and Asia. Plane trees are planted widely in cities for their resistance to diseases and to air pollution and because they grow rapidly and furnish quick shade. They are characterized by scaling bark; large, deciduous, usually lobed leaves; and globular heads of flower and seed. Ball-shaped smooth or bristly seed clusters, which dangle singly and often persist after leaf fall, are key identifiers. Winter bark is patchy and picturesque; as the outer bark flakes off, inner bark shows shades of white, gray, green, and yellow.

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or bo tree

In Buddhism, the fig tree under which the Buddha sat when he attained enlightenment (bodhi) at Bodh Gaya (near Gaya, India). The tree growing on the site now is believed to be a descendant of the original, planted from a cutting of a tree in Sri Lanka that had been propagated from the original; both trees are sites of pilgrimage for Buddhists. The bo tree or a representation of its leaf has often been used as a symbol of the Buddha.

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Diagram showing the evolutionary interrelations of a group of organisms that usually originated from a shared ancestral form. The ancestor is in the tree trunk; organisms that have arisen from it are placed at the ends of tree branches. The distance of one group from the other groups indicates the degree of relationship; that is, closely related groups are located on branches close to one another. Though phylogenetic trees are speculative, they provide a convenient method for studying phylogenetic relationships and evolution. Seealso phylogeny.

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or garden pepper

Red peppers (Capsicum annuum) from which paprika is made

Any of many plants in the genus Capsicum of the nightshade family, notably C. annuum, C. frutescens, and C. boccatum, native to Central and South America and cultivated extensively throughout tropical Asia and the equatorial New World for their edible, pungent fruits. Red, green, and yellow mild bell or sweet peppers, rich in vitamins A and C, are used in seasoning and as a vegetable food. The pungency of hot peppers, including tabasco, chili, and cayenne peppers, comes from the compound capsaicin in the internal partitions of the fruit. The spice black pepper comes from an unrelated plant.

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or pepper

Perennial, woody climbing vine (Piper nigrum) of the family Piperaceae, native to India; also, the hotly pungent spice made from its berries. One of the earliest spices known, pepper is probably the most widely used spice in the world today. It early became an important article of overland trade between India and Europe. The plant is cultivated throughout Indonesia and has been introduced into tropical areas elsewhere. It has broad, shiny leaves and dense, slender spikes of small flowers. The small berrylike fruits are called peppercorns. Seealso pepper.

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(born Sept. 8, 1900, Dudleyville, Ala., U.S.—died May 30, 1989, Washington, D.C.) U.S. politician. He practiced law in Florida before being elected to the U.S. Senate (1937–51), where he supported legislation that created Social Security, a minimum wage, and medical assistance for the elderly. Elected to the U.S. House of Representatives (1963–89), he chaired the committee on aging and sponsored legislation abolishing mandatory retirement in federal agencies and raising the retirement age to 70 in the private sector (1968). In 1989 he was awarded the Medal of Freedom.

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(born Sept. 8, 1900, Dudleyville, Ala., U.S.—died May 30, 1989, Washington, D.C.) U.S. politician. He practiced law in Florida before being elected to the U.S. Senate (1937–51), where he supported legislation that created Social Security, a minimum wage, and medical assistance for the elderly. Elected to the U.S. House of Representatives (1963–89), he chaired the committee on aging and sponsored legislation abolishing mandatory retirement in federal agencies and raising the retirement age to 70 in the private sector (1968). In 1989 he was awarded the Medal of Freedom.

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Monkey puzzle tree (Araucaria araucana).

Evergreen ornamental and timber conifer (Araucaria araucana) of the family Araucariaceae, native to the Andes Mountains of South America. The tree may grow to 150 ft (45 m) in height and 5 ft (1.5 m) in diameter. The rigid, overlapping, needle-pointed leaves are spirally arranged on stiff branches, which form a tangled, prickly network that discourages animals from climbing the tree. The Norfolk Island pine is a relative.

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Ginkgo (Ginkgo biloba)

Tree (Ginkgo biloba, family Ginkgoaceae) that is the only living representative of the gymnosperm order Ginkgoales. Native to China, it is often termed a living fossil because it is unclear whether uncultivated groups can be found in the wild. It has been planted since ancient times in Chinese and Japanese temple gardens and is now valued in many parts of the world as an attractive, fungus- and insect-resistant ornamental tree. It tolerates cold weather and, unlike most gymnosperms, can survive the adverse atmospheric conditions of urban areas. Pyramidal in shape, it has a columnar, sparingly branched trunk. The light-coloured wood, soft and weak, has little economic value. The fan-shaped, leathery leaves, most divided into two lobes by a central notch, resemble the leaflets of the maidenhair fern. The silvery nut, when roasted, is considered a delicacy. Studies have suggested that Ginkgo biloba supplements can enhance memory function in the elderly and delay the onset of Alzheimer disease.

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Any of the more than 500 species of mostly very large trees in the genus Eucalyptus, in the myrtle family, native to Australia, New Zealand, Tasmania, and nearby islands. Many species are grown widely throughout the temperate regions of the world as shade trees or in forestry plantations. Because they grow rapidly, many species attain great height. The leaf glands of many species, especially E. salicifolia and E. globulus, contain a volatile, aromatic oil known as eucalyptus oil, used mostly in medicines. Eucalyptus wood is used extensively in Australia as fuel, and the timber is commonly used in buildings and fencing. The bark of many species is used in papermaking and tanning.

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Broad-leaved tree that sheds all its leaves during one season. Deciduous forests are found in three middle-latitude regions with a temperate climate characterized by a winter season and year-round precipitation: eastern North America, western Eurasia, and northeastern Asia. They also extend into more arid regions along stream banks and around bodies of water. Oaks, beeches, birches, chestnuts, aspens, elms, maples, and basswoods (or lindens) are the dominant trees in mid-latitude deciduous forests. Other plants that shed their leaves seasonally may also be called deciduous. Seealso conifer, evergreen.

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Tree (Crescentia cujete) of the trumpet-creeper family (Bignoniaceae) that grows in Central and South America, the West Indies, and extreme southern Florida. It is often grown as an ornamental. It produces large spherical fruits, the hard shells of which are useful as bowls, cups, and other water containers when hollowed out. The fruit's shell encloses a whitish pulp and thin, dark brown seeds. The tree bears funnel-shaped, light green and purple-streaked flowers and evergreen leaves. Fruits of the unrelated bottle gourd (Lagenaria siceraria) are also known as calabashes.

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or giant sequoia or Sierra redwood

Coniferous evergreen (Sequoiadendron giganteum; see conifer) found in scattered groves on the western slopes of the Sierra Nevada range of California, U.S. The largest of all trees in bulk, the big tree is distinguished from the coastal redwood (Sequoia sempervirens) by having uniformly scalelike, or awl-shaped, leaves that lie close against the branches, scaleless winter buds, and cones requiring two seasons to mature. The pyramidal tree shape, reddish brown furrowed bark, and drooping branches are common to both genera. The largest specimen (in total bulk) is the General Sherman tree in Sequoia National Park—101.5 ft (31 m) in circumference at its base, 272.4 ft (83 m) tall, and weighing an estimated 6,167 tons (5,593 metric tons). Because big-tree lumber is more brittle than redwood lumber and thus less desirable, the big tree has been easier to preserve; though some groves have been cut, most of the 70 remaining groves are now protected by state or national forests or parks.

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(born Dec. 17, 1853, London, Eng.—died July 2, 1917, London) British actor-manager. A romantic actor with a gift for character roles and comedy, he made his London debut in 1878 and won favourable notice in The Private Secretary (1884). As manager of the Haymarket Theatre (1887–97), he directed and acted in lavish Shakespearean productions, which he continued as actor-manager of Her Majesty's Theatre (1897–1915). He produced notable stage versions of Charles Dickens's works. In 1904 he founded the Royal Academy of Dramatic Art.

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(born Dec. 17, 1853, London, Eng.—died July 2, 1917, London) British actor-manager. A romantic actor with a gift for character roles and comedy, he made his London debut in 1878 and won favourable notice in The Private Secretary (1884). As manager of the Haymarket Theatre (1887–97), he directed and acted in lavish Shakespearean productions, which he continued as actor-manager of Her Majesty's Theatre (1897–1915). He produced notable stage versions of Charles Dickens's works. In 1904 he founded the Royal Academy of Dramatic Art.

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National park, southeastern California, U.S. Situated on the border between the Mojave and Colorado deserts, it has an area of 1,241 sq mi (3,214 sq km). It was designated a national monument in 1936 and a national park in 1994. It is noted for its variety of desert plant life, including the Joshua tree, creosote bush, and Mojave yucca. Its fauna include coyotes, bobcats, and tarantulas.

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In computer science, 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.

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.

A B-tree is kept balanced by requiring that all external 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 257-513 B-tree.

A B-tree of order m (the maximum number of children for each node) is a tree which satisfies the following properties :

  1. Every node has <= m children.
  2. Every node (except root and leaves) has >= m/2 children.
  3. The root has at least 2 children if it is not a leaf node.
  4. All leaves appear in the same level, and carry no information.
  5. A non-leaf node with k children contains k – 1 keys

The B-tree's creators, Rudolf Bayer and Ed McCreight, have not explained what, if anything, the B stands for. The most common belief is that B stands for balanced, as all the leaf nodes are at the same level in the tree. B may also stand for Bayer, Branching Tree, or for Boeing, because they were working for Boeing Scientific Research Labs at the time.

Node structures

Each internal node's elements act as separation values which divide its subtrees. For example, if an internal node has three child nodes (or subtrees) then it must have two separation values or elements a1 and a2. All values in the leftmost subtree will be less than a1 , all values in the middle subtree will be between a1 and a2, and all values in the rightmost subtree will be greater than a2.

Internal nodes in a B-tree — nodes which are not leaf nodes — are usually represented as an ordered set of elements and child pointers. Every internal node contains a maximum of U children and — other than the root — a minimum of L children. For all internal nodes other than the root, the number of elements is one less than the number of child pointers; the number of elements is between L-1 and U-1. The number U must be either 2L or 2L-1; thus each internal node is at least half full. This relationship between U and L implies that two half-full nodes can be joined to make a legal node, and one full node can be split into two legal nodes (if there is room to push one element up into the parent). These properties make it possible to delete and insert new values into a B-tree and adjust the tree to preserve the B-tree properties.

Leaf nodes have the same restriction on the number of elements, but have no children, and no child pointers.

The root node still has the upper limit on the number of children, but has no lower limit. For example, when there are fewer than L-1 elements in the entire tree, the root will be the only node in the tree, and it will have no children at all.

A B-tree of depth n+1 can hold about U times as many items as a B-tree of depth n, but the cost of search, insert, and delete operations grows with the depth of the tree. As with any balanced tree, the cost grows much more slowly than the number of elements.

Some balanced trees store values only at the leaf nodes, and so have different kinds of nodes for leaf nodes and internal nodes. B-trees keep values in every node in the tree, and may use the same structure for all nodes. However, since leaf nodes never have children, a specialized structure for leaf nodes in B-trees will improve performance.

Algorithms

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.

Insertion

All insertions happen at the leaf nodes.

  1. By searching the tree, find the leaf node where the new element should be added.
  2. If the leaf node contains fewer than the maximum legal number of elements, there is room for one more. Insert the new element in the node, keeping the node's elements ordered.
  3. Otherwise the leaf node is split into two nodes.
    1. A single median is chosen from among the leaf's elements and the new element.
    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. That separation value is added to the node's parent, which may cause it to be split, and so on.

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 too is legal. 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 improved 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. 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 improved 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.

Deletion

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

  • locate and delete the item, then restructure the tree to regain its invariants

or

  • 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 may be a separator for its child nodes
  2. deleting an element may put it under the minimum number of elements and children.

Each of these cases will be dealt with in order.

Deletion from a leaf node

  • Search for the value to delete.
  • If the value is in a leaf node, it can simply be deleted from the node, perhaps leaving the node with too few elements; so some additional changes to the tree will be required.

Deletion from an internal node

Each element in an internal node acts as a separation value for two subtrees, and when such an element is deleted, two cases arise. In the first case, both of the two child nodes to the left and right of the deleted element have the minimum number of elements, namely L-1. They can then be joined into a single node with 2L-2 elements, a number which does not exceed U-1 and so is a legal node. Unless it is known that this particular B-tree does not contain duplicate data, we must then also (recursively) delete the element in question from the new node.

In the second case, one of the two child nodes contains more than the minimum number of elements. Then a new separator for those subtrees must be found. Note that the largest element in the left subtree is the largest element which is still less than the separator. Likewise, the smallest element in the right subtree is the smallest element which is still greater than the separator. Both of those elements are in leaf nodes, and either can be the new separator for the two subtrees.

  • If the value is in an internal node, 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.
  • This has deleted an element from a leaf node, and so is now equivalent to the previous case.

Rebalancing after deletion

If deleting an element from a leaf node has brought it under the minimum size, some elements must be redistributed to bring all nodes up to the minimum. In some cases the rearrangement will move the deficiency to the parent, and the redistribution must be applied iteratively up the tree, perhaps even 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 right sibling has more than the minimum number of elements.
    • Add the separator to the end of the deficient node.
    • Replace the separator in the parent with the first element of the right sibling.
    • Make the first child of the right sibling into the last child of the deficient node
  • Otherwise, if the left sibling has more than the minimum number of elements.
    • Add the separator to the start of the deficient node.
    • Replace the separator in the parent with the last element of the left sibling.
    • Make the last child of the left sibling into the first child of the deficient node
  • If both immediate siblings have only the minimum number of elements
    • Create a new node with all the elements from the deficient node, all the elements from one of its siblings, and the separator in the parent between the two combined sibling nodes.
    • Remove the separator from the parent, and replace the two children it separated with the combined node.
    • If that brings the number of elements in the parent under the minimum, repeat these steps with that deficient node, unless it is the root, since the root may be deficient.

The only other case to account for is when the root has no elements and one child. In this case it is sufficient to replace it with its only child.

Creating a B-tree

When a large amount of information needs to be put into B-tree form, it might be assumed that simply repetitively inserting the data would be the quickest. In practice this is a slow operation due to the rearrangement of the tree that occurs. It is much quicker to sort the data first, and then insert it as a monolithic block.

Notes

Each node will always have between L and U children, inclusively, with one exception: the root node may have anywhere from 2 to U children inclusively. In other words, the root is exempt from the lower bound restriction. This allows the tree to hold small numbers of elements. The root having one child makes no sense, since the subtree attached to that child could simply be attached to the root. Giving the root no children is also unnecessary, since a tree with no elements is typically represented as having no root node.

Multi-way combining and splitting

It is possible to modify the above algorithm to, when trying to find extra elements for a deficient node, examine other siblings, and if one has more than the minimum number of values rearrange values across a larger number of siblings to make up the deficit in one.

Similarly, when a node is split, extra elements can be moved to nearby, less populated siblings; or the split can involve a number of siblings, redistributing elements among them rather than splitting a node.

In practice, the most common use of B-trees involves keeping the nodes on secondary storage, where it is slow to access a node which is not already being used. Using only two-ways splits and combines helps decrease the number of nodes needed for many common situations, but may be useful in others.

Relationship between U and L

It is almost universal to split nodes by choosing a single median and creating two new nodes. This constrains the relationship between L and U. Trying to insert an element into a node with U elements — involves redistributing U elements. One of these, the median, will move to the parent, and the remaining elements will be split as equally as possible among the two new nodes.

For example, in a 2-3 B-tree, adding an element to a node with three child nodes, and thus two separator values, involves three values — the two separators and the new value. The median becomes the new separator in the parent, and each of the other two becomes the sole elements in nodes with one value and two children. Generally, if U is odd, each of the two new nodes has (U+1)/2 children. If U is even, one has U/2 children and the other U/2+1.

If full nodes are split into exactly two nodes, L must be small enough to allow for the sizes after a node is split. But it is possible to split full nodes into more than two new nodes. Choosing to split a node into more than two nodes would require a lower value of L for the same value of U.

As L gets smaller, it allows for more unused space in the nodes. This might decrease the frequency of node splitting, but it is also likely to increase the amount of memory needed to store the same number of values, and the number of nodes that have to be examined for any particular operation.

Theoretical results

Robert Tarjan proved that the amortized number of splits/merges is 2.

Access Concurrency

Lehman and Yao showed that linking the tree blocks at each level together with a next pointer results in a tree structure where read locks on the tree blocks can be avoided as the tree is descended from the root to the leaf for both search and insertion. Write locks are only required as a tree block is modified. Minimizing locking to a single node held only during its modification helps to maximize access concurrency by multiple users, an important consideration for databases and/or other B-Tree based ISAM storage methods.

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