Incomputer science, asearch tree is atree data structure used for locating specific keys from within a set. In order for a tree to function as a search tree, the key for each node must be greater than any keys in subtrees on the left, and less than any keys in subtrees on the right.[1]
The advantage of search trees is their efficient search time given the tree is reasonably balanced, which is to say theleaves at either end are of comparable depths. Various search-tree data structures exist, several of which also allow efficient insertion and deletion of elements, which operations then have to maintain tree balance.
Search trees are often used to implement anassociative array. The search tree algorithm uses the key from thekey–value pair to find a location, and then the application stores the entire key–value pair at that particular location.
A Binary Search Tree is a node-based data structure where each node contains a key and two subtrees, the left and right. For all nodes, the left subtree's key must be less than the node's key, and the right subtree's key must be greater than the node's key. These subtrees must all qualify as binary search trees.
The worst-casetime complexity for searching a binary search tree is theheight of the tree, which can be as small as O(log n) for a tree with n elements.
B-trees are generalizations of binary search trees in that they can have a variable number of subtrees at each node. While child-nodes have a pre-defined range, they will not necessarily be filled with data, meaning B-trees can potentially waste some space. The advantage is that B-trees do not need to be re-balanced as frequently as otherself-balancing trees.
Due to the variable range of their node length, B-trees are optimized for systems that read large blocks of data, they are also commonly used in databases.
The time complexity for searching a B-tree is O(log n).
An (a,b)-tree is a search tree where all of its leaves are the same depth. Each node has at leasta children and at mostb children, while the root has at least 2 children and at mostb children.
a andb can be decided with the following formula:[2]
The time complexity for searching an (a,b)-tree is O(log n).
A ternary search tree is a type oftree that can have 3 nodes: a low child, an equal child, and a high child. Each node stores a single character and the tree itself is ordered the same way a binary search tree is, with the exception of a possible third node.
Searching a ternary search tree involves passing in astring to test whether any path contains it.
The time complexity for searching a balanced ternary search tree is O(log n).
Assuming the tree is ordered, we can take a key and attempt to locate it within the tree. The following algorithms are generalized for binary search trees, but the same idea can be applied to trees of other formats.
search-recursive(key, node)if node isNULLreturnEMPTY_TREEif key < node.key return search-recursive(key, node.left)else if key > node.key return search-recursive(key, node.right)elsereturn node
searchIterative(key, node) currentNode := nodewhile currentNode is notNULLif currentNode.key = keyreturn currentNodeelse if currentNode.key > key currentNode := currentNode.leftelse currentNode := currentNode.right
In a sorted tree, the minimum is located at the node farthest left, while the maximum is located at the node farthest right.[3]
findMinimum(node)if node isNULLreturnEMPTY_TREE min := nodewhile min.left is notNULL min := min.leftreturn min.key
findMaximum(node)if node isNULLreturnEMPTY_TREE max := nodewhile max.right is notNULL max := max.rightreturn max.key
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