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|Worst-case performance||O(n²) (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.
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(n²) time for this sorting algorithm. This worst case occurs when the algorithm operates on an already sorted set, or one that is nearly sorted, reversed or nearly reversed. Expected O(n log n) time can however be achieved by shuffling the array, but this does not help for equal items.
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, tree sort algorithms require separate memory to be allocated for the tree, as opposed to in-place algorithms such as quicksort or heapsort. On most common platforms, this means that heap memory has to be used, 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.
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)
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(n²) worst-case scenarios.
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