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Depending on the problem at hand, pre-order, post-order, and especially one of the number of subtrees − 1 in-order operations may be optional. Also, in practice more than one of pre-order, post-order, and in-order operations may be required. For example, when inserting into a ternary tree, a pre-order operation is performed by comparing items.
The necessary distinction can be made by first partitioning the edges; i.e., defining the binary tree as triplet (V, E 1, E 2), where (V, E 1 ∪ E 2) is a rooted tree (equivalently arborescence) and E 1 ∩ E 2 is empty, and also requiring that for all j ∈ { 1, 2 }, every node has at most one E j child. [14]
A tree sort is a sort algorithm that builds a binary search tree from the elements to be sorted, and then traverses the tree so that the elements come out in sorted order. [1]
The name preorder is meant to suggest that preorders are almost partial orders, but not quite, as they are not necessarily antisymmetric. A natural example of a preorder is the divides relation "x divides y" between integers, polynomials, or elements of a commutative ring. For example, the divides relation is reflexive as every integer divides ...
Encompassment is a preorder, i.e. reflexive and transitive, but not anti-symmetric, [note 1] nor total [note 2] The corresponding equivalence relation, defined by s ~ t if s ≤ t ≤ s, is equality modulo renaming. s ≤ t whenever s is a subterm of t. s ≤ t whenever t is a substitution instance of s.
One problem with this algorithm is that, because of its recursion, it uses stack space proportional to the height of a tree. If the tree is fairly balanced, this amounts to O(log n) space for a tree containing n elements. In the worst case, when the tree takes the form of a chain, the height of the tree is n so the algorithm takes O(n) space. A ...
To turn a regular search tree into an order statistic tree, the nodes of the tree need to store one additional value, which is the size of the subtree rooted at that node (i.e., the number of nodes below it).
[8]: ch. 8 In order for search to work effectively it has to employ a comparison function which establishes a total order (or at least a total preorder) on the set of keys. [9]: 23 The number of comparisons required for successful search is limited by the height h and for unsuccessful search is very close to h, so both are in O(log n).