Tree sort

Tree sort

Class	Sorting algorithm
Data structure	Array
Worst-case performance	$O (n ²)$ (unbalanced) $O (n log n)$ (balanced)
Best-case performance	$O (n log n)$
Average performance	$O (n log n$ )
Worst-case space complexity	$Θ(n)$

A tree sort is a sort algorithm that builds a binary search tree from the elements to be sorted, and then traverses the tree (in-order) so that the elements come out in sorted order. Its typical use is sorting elements online: after each insertion, the set of elements seen so far is available in sorted order.

Efficiency

Adding one item to a binary search tree is on average an $O (log n)$ process (in big O notation). Adding n items is an $O (n log n)$ process, making tree sorting a 'fast sort' process. Adding an item to an unbalanced binary tree requires $O (n)$ time in the worst-case: When the tree resembles a linked list (degenerate tree). This results in a worst case of $O (n ²)$ time for this sorting algorithm. This worst case occurs when the algorithm operates on an already sorted set, or one that is nearly sorted. Expected $O (n log n)$ time can however be achieved by shuffling the array.

The worst-case behaviour can be improved by using a self-balancing binary search tree. Using such a tree, the algorithm has an $O (n log n)$ worst-case performance, thus being degree-optimal for a comparison sort. However, trees require memory to be allocated on the heap, which is a significant performance hit when compared to quicksort and heapsort. When using a splay tree as the binary search tree, the resulting algorithm (called splaysort) has the additional property that it is an adaptive sort, meaning that its running time is faster than $O (n log n)$ for inputs that are nearly sorted.

Example

The following tree sort algorithm in pseudocode accepts a collection of comparable items and outputs the items in ascending order:

STRUCTURE BinaryTree
    BinaryTree:LeftSubTree
    Object:Node
    BinaryTree:RightSubTree

PROCEDURE Insert(BinaryTree:searchTree, Object:item)
    IF searchTree.Node IS NULL THEN
        SET searchTree.Node TO item
    ELSE
        IF item IS LESS THAN searchTree.Node THEN
            Insert(searchTree.LeftSubTree, item)
        ELSE
            Insert(searchTree.RightSubTree, item)

PROCEDURE InOrder(BinaryTree:searchTree)
    IF searchTree.Node IS NULL THEN
        EXIT PROCEDURE
    ELSE
        InOrder(searchTree.LeftSubTree)
        EMIT searchTree.Node
        InOrder(searchTree.RightSubTree)

PROCEDURE TreeSort(Collection:items)
    BinaryTree:searchTree
   
    FOR EACH individualItem IN items
        Insert(searchTree, individualItem)
   
    InOrder(searchTree)

In a simple functional programming form, the algorithm (in Haskell) would look something like this:

data Tree a = Leaf | Node (Tree a) a (Tree a)

insert :: Ord a => a -> Tree a -> Tree a
insert x Leaf = Node Leaf x Leaf
insert x (Node t y s)
    | x <= y = Node (insert x t) y s
    | x > y  = Node t y (insert x s)

flatten :: Tree a -> [a]
flatten Leaf = []
flatten (Node t x s) = flatten t ++ [x] ++ flatten s

treesort :: Ord a => [a] -> [a]
treesort = flatten . foldr insert Leaf

In the above implementation, both the insertion algorithm and the retrieval algorithm have $O (n ²)$ worst-case scenarios.

External links

The Wikibook Algorithm Implementation has a page on the topic of: Binary Tree Sort

Sorting algorithms
Theory	Computational complexity theory Big O notation Total order Lists Inplacement Stability Comparison sort Adaptive sort Sorting network Integer sorting X + Y sorting Transdichotomous model Quantum sort
Exchange sorts	Bubble sort Cocktail shaker sort Odd–even sort Comb sort Gnome sort Quicksort Slowsort Stooge sort Bogosort
Selection sorts	Selection sort Heapsort Smoothsort Cartesian tree sort Tournament sort Cycle sort Weak-heap sort
Insertion sorts	Insertion sort Shellsort Splaysort Tree sort Library sort Patience sorting
Merge sorts	Merge sort Cascade merge sort Oscillating merge sort Polyphase merge sort
Distribution sorts	American flag sort Bead sort Bucket sort Burstsort Counting sort Pigeonhole sort Proxmap sort Radix sort Flashsort
Concurrent sorts	Bitonic sorter Batcher odd–even mergesort Pairwise sorting network
Hybrid sorts	Block merge sort Timsort Introsort Spreadsort
Other	Topological sorting Pancake sorting Spaghetti sort

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