Tree traversal: Difference between revisions

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==Types==

{{Tree traversal demo}}

Unlike [[linked list]]s, [[one-dimensional array]]s and other [[List of data structures#Linear data structures|linear data structures]], which are canonically traversed in linear order, trees may be traversed in multiple ways. They may be traversed in [[Depth-first search|depth-first]] or [[Breadth-first search|breadth-first]] order. There are three common ways to traverse them in depth-first order: in-order, pre-order and post-order.<ref name="holtenotes">{{cite web|url=http://webdocs.cs.ualberta.ca/~holte/T26/tree-traversal.html|title=Lecture 8, Tree Traversal|access-date=2 May 2015}}</ref> Beyond these basic traversals, various more complex or hybrid schemes are possible, such as [[depth-limited search]]es like [[iterative deepening depth-first search]]. The latter, as well as breadth-first search, can also be used to traverse infinite trees, see [[#Infinite trees|below]].

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# Visit the current node (in the figure: position blue).

Post-order traversal can be useful to get [[Reverse_Polish_notation|postfix expression]] of a [[binary expression tree]].

===={{anchor|Inorder traversal|In-order traversal}}In-order, LNR====

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==Applications==

[[File:AST binary tree arith variables.svg|260px|thumb|Tree representing the arithmetic expression: ''A'' * {{nowrap|(''B'' − ''C'')}} + {{nowrap|(''D'' + ''E'')}}]]

Pre-order traversal can be used to make a prefix expression ([[Polish notation]]) from [[Parse tree|expression trees]]: traverse the expression tree pre-orderly. For example, traversing the depicted arithmetic expression in pre-order yields "+ * ''A'' − ''B'' ''C'' + ''D'' ''E''". In prefix notation, there is no need for any parentheses as long as each operator has a fixed number of operands. PreorderPre-order traversal is also used to create a copy of the tree.

Post-order traversal can generate a postfix representation ([[Reverse Polish notation]]) of a binary tree. Traversing the depicted arithmetic expression in post-order yields "''A'' ''B'' ''C'' − * ''D'' ''E'' + +"; the latter can easily be transformed into [[machine code]] to evaluate the expression by a [[stack machine]]. PostorderPost-order traversal is also used to delete the tree. Each node is freed after freeing its children.

In-order traversal is very commonly used on [[binary search tree]]s because it returns values from the underlying set in order, according to the comparator that set up the binary search tree.

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{{unreferenced section|date=June 2013}}

===Depth-first search implementation===

Below are examples of [[stack (abstract data type)|stack]]-based implementation for pre-order, post-order and in-order traversal in recursive approach (left) as well as iterative approach (right).

Implementations in iterative approach are able to avoid the [[Recursion (computer science)#Recursion versus iteration|drawbacks of recursion]], particularly limitations of stack space and performance issues.

Several alternative implementations are also mentioned.

===={{anchor|Pre-order traversal code|Pre-order traversal code}}Pre-order implementation====

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|

'''procedure''' iterativePostorder(node)

'''if''' node = '''null'''

'''return'''

stack ← '''empty stack'''

lastNodeVisited ← '''null'''

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|

'''procedure''' iterativeInorder(node)

'''if''' node = '''null'''

'''return'''

stack ← '''empty stack'''

'''while''' '''not''' stack.isEmpty() '''or''' node ≠ '''null'''

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|}

====Another variant of Prepre-order====

If the tree is represented by an array (first index is 0), it is possible to calculate the index of the next element:<ref>{{Cite web|title=constexpr tree structures|url=https://fekir.info/post/constexpr-tree/#_dfs_traversal|access-date=2021-08-15|website=Fekir's Blog|date=9 August 2021|language=en}}</ref>{{clarify|reason=Explicitly mention the restrictions on trees in order to be handled by this algorithm. Since there is no isLeaf() test, it seems that all leaves must be on maximal depth or one level above it, like in a [[heap (data structure)]].|date=November 2021}}

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Thus, simple depth-first or breadth-first searches do not traverse every infinite tree, and are not efficient on very large trees. However, hybrid methods can traverse any (countably) infinite tree, essentially via a [[Diagonal argument (disambiguation)|diagonal argument]] ("diagonal"—a combination of vertical and horizontal—corresponds to a combination of depth and breadth).

Concretely, given the infinitely branching tree of infinite depth, label the root (), the children of the root (1), (2), …..., the grandchildren (1, 1), (1, 2), …..., (2, 1), (2, 2), …..., and so on. The nodes are thus in a [[bijection|one-to-one]] correspondence with finite (possibly empty) sequences of positive numbers, which are countable and can be placed in order first by sum of entries, and then by [[lexicographic order]] within a given sum (only finitely many sequences sum to a given value, so all entries are reached—formally there are a finite number of [[Composition (number theory)|compositions]] of a given natural number, specifically 2^''n''−1 compositions of {{nowrap|1=''n'' ≥ 1}}), which gives a traversal. Explicitly:

# ()

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* [http://rosettacode.org/wiki/Tree_traversal See tree traversal implemented in various programming language] on [[Rosetta Code]]

* [http://www.perlmonks.org/?node_id=600456 Tree traversal without recursion]

* [https://www.geeksforgeeks.org/tree-traversals-inorder-preorder-and-postorder/ Tree Traversal Algorithms]

* [https://faculty.cs.niu.edu/~mcmahon/CS241/Notes/Data_Structures/binary_tree_traversals.html Binary Tree Traversal]

* [https://www.simplilearn.com/tutorials/data-structure-tutorial/tree-traversal-in-data-structure Tree Traversal In Data Structure]

{{Graph traversal algorithms}}