Graph and tree search algorithms 


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In computer science, tree traversal (also known as tree search and walking the tree) is a form of graph traversal and refers to the process of visiting (checking and/or updating) each node in a tree data structure, exactly once. Such traversals are classified by the order in which the nodes are visited. The following algorithms are described for a binary tree, but they may be generalized to other trees as well.
Unlike linked lists, onedimensional arrays and other linear data structures, which are canonically traversed in linear order, trees may be traversed in multiple ways. They may be traversed in depthfirst or breadthfirst order. There are three common ways to traverse them in depthfirst order: inorder, preorder and postorder.^{[1]} Beyond these basic traversals, various more complex or hybrid schemes are possible, such as depthlimited searches like iterative deepening depthfirst search. The latter, as well as breadthfirst search, can also be used to traverse infinite trees, see below.
Traversing a tree involves iterating over all nodes in some manner. Because from a given node there is more than one possible next node (it is not a linear data structure), then, assuming sequential computation (not parallel), some nodes must be deferred—stored in some way for later visiting. This is often done via a stack (LIFO) or queue (FIFO). As a tree is a selfreferential (recursively defined) data structure, traversal can be defined by recursion or, more subtly, corecursion, in a very natural and clear fashion; in these cases the deferred nodes are stored implicitly in the call stack.
Depthfirst search is easily implemented via a stack, including recursively (via the call stack), while breadthfirst search is easily implemented via a queue, including corecursively.^{[2]}^{:45−61}
These searches are referred to as depthfirst search (DFS), since the search tree is deepened as much as possible on each child before going to the next sibling. For a binary tree, they are defined as access operations at each node, starting with the current node, whose algorithm is as follows:^{[3]}^{[4]}
The general recursive pattern for traversing a binary tree is this:
There are three methods (patterns) at which position of the parcours (traversal) relative to the node (in the figure: red, green, or blue) the visit (access) of the node shall take place. The choice of exactly one color determines exactly one visit of a node as described below. Access at all three colors results in a threefold visit of the same node yielding the “allorder” sequentialisation:
The preorder traversal is a topologically sorted one, because a parent node is processed before any of its child nodes is done.
In a binary search tree ordered such that in each node the key is greater than all keys in its left subtree and less than all keys in its right subtree, inorder traversal retrieves the keys in ascending sorted order.^{[6]}
In a binary search tree, reverse inorder traversal retrieves the keys in descending sorted order.
The trace of a traversal is called a sequentialisation of the tree. The traversal trace is a list of each visited node. No one sequentialisation according to pre, in or postorder describes the underlying tree uniquely. Given a tree with distinct elements, either preorder or postorder paired with inorder is sufficient to describe the tree uniquely. However, preorder with postorder leaves some ambiguity in the tree structure.^{[7]}
To traverse any tree with depthfirst search, perform the following operations at each node:
Depending on the problem at hand, the preorder, postorder, and especially one of the (number_of_children − 1) inorder operations may be void, or access only to a specific child is wanted, so these operations are optional. Also, in practice more than one of preorder, inorder and postorder operations may be required. For example, when inserting into a ternary tree, a preorder operation is performed by comparing items. A postorder operation may be needed afterwards to rebalance the tree.
Trees can also be traversed in levelorder, where we visit every node on a level before going to a lower level. This search is referred to as breadthfirst search (BFS), as the search tree is broadened as much as possible on each depth before going to the next depth.
There are also tree traversal algorithms that classify as neither depthfirst search nor breadthfirst search. One such algorithm is Monte Carlo tree search, which concentrates on analyzing the most promising moves, basing the expansion of the search tree on random sampling of the search space.
Preorder traversal can be used to make a prefix expression (Polish notation) from expression trees: traverse the expression tree preorderly. For example, traversing the depicted arithmetic expression in preorder yields "+ * A − B C + D E".
Postorder traversal can generate a postfix representation (Reverse Polish notation) of a binary tree. Traversing the depicted arithmetic expression in postorder yields "A B C − * D E + +"; the latter can easily be transformed into machine code to evaluate the expression by a stack machine.
Inorder traversal is very commonly used on binary search trees because it returns values from the underlying set in order, according to the comparator that set up the binary search tree.
Postorder traversal while deleting or freeing nodes and values can delete or free an entire binary tree. Thereby the node is freed after freeing its children.
Also the duplication of a binary tree yields a postorder sequence of actions, because the pointer copy to the copy of a node is assigned to the corresponding child field N.child within the copy of the parent N immediately after return
copy in the recursive procedure. This means that the parent cannot be finished before all children are finished.
This section does not cite any sources.(June 2013) 
preorder(node) if (node == null) return visit(node) preorder(node.left) preorder(node.right) 
iterativePreorder(node) if (node == null) return s ← empty stack s.push(node) while (not s.isEmpty()) node ← s.pop() visit(node) //right child is pushed first so that left is processed first if node.right ≠ null s.push(node.right) if node.left ≠ null s.push(node.left) 
inorder(node) if (node == null) return inorder(node.left) visit(node) inorder(node.right) 
iterativeInorder(node) s ← empty stack while (not s.isEmpty() or node ≠ null) if (node ≠ null) s.push(node) node ← node.left else node ← s.pop() visit(node) node ← node.right 
postorder(node) if (node == null) return postorder(node.left) postorder(node.right) visit(node) 
iterativePostorder(node) s ← empty stack lastNodeVisited ← null while (not s.isEmpty() or node ≠ null) if (node ≠ null) s.push(node) node ← node.left else peekNode ← s.peek() // if right child exists and traversing node // from left child, then move right if (peekNode.right ≠ null and lastNodeVisited ≠ peekNode.right) node ← peekNode.right else visit(peekNode) lastNodeVisited ← s.pop() 
All the above implementations require stack space proportional to the height of the tree which is a call stack for the recursive and a parent stack for the iterative ones. In a poorly balanced tree, this can be considerable. With the iterative implementations we can remove the stack requirement by maintaining parent pointers in each node, or by threading the tree (next section).
A binary tree is threaded by making every left child pointer (that would otherwise be null) point to the inorder predecessor of the node (if it exists) and every right child pointer (that would otherwise be null) point to the inorder successor of the node (if it exists).
Advantages:
Disadvantages:
Morris traversal is an implementation of inorder traversal that uses threading:^{[8]}
Also, listed below is pseudocode for a simple queue based levelorder traversal, and will require space proportional to the maximum number of nodes at a given depth. This can be as much as the total number of nodes / 2. A more spaceefficient approach for this type of traversal can be implemented using an iterative deepening depthfirst search.
levelorder(root) q ← empty queue q.enqueue(root) while not q.isEmpty() do node ← q.dequeue() visit(node) if node.left ≠ null then q.enqueue(node.left) if node.right ≠ null then q.enqueue(node.right)
While traversal is usually done for trees with a finite number of nodes (and hence finite depth and finite branching factor) it can also be done for infinite trees. This is of particular interest in functional programming (particularly with lazy evaluation), as infinite data structures can often be easily defined and worked with, though they are not (strictly) evaluated, as this would take infinite time. Some finite trees are too large to represent explicitly, such as the game tree for chess or go, and so it is useful to analyze them as if they were infinite.
A basic requirement for traversal is to visit every node eventually. For infinite trees, simple algorithms often fail this. For example, given a binary tree of infinite depth, a depthfirst search will go down one side (by convention the left side) of the tree, never visiting the rest, and indeed an inorder or postorder traversal will never visit any nodes, as it has not reached a leaf (and in fact never will). By contrast, a breadthfirst (levelorder) traversal will traverse a binary tree of infinite depth without problem, and indeed will traverse any tree with bounded branching factor.
On the other hand, given a tree of depth 2, where the root has infinitely many children, and each of these children has two children, a depthfirst search will visit all nodes, as once it exhausts the grandchildren (children of children of one node), it will move on to the next (assuming it is not postorder, in which case it never reaches the root). By contrast, a breadthfirst search will never reach the grandchildren, as it seeks to exhaust the children first.
A more sophisticated analysis of running time can be given via infinite ordinal numbers; for example, the breadthfirst search of the depth 2 tree above will take ω·2 steps: ω for the first level, and then another ω for the second level.
Thus, simple depthfirst or breadthfirst 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 ("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 onetoone 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 compositions of a given natural number, specifically 2^{n−1} compositions of n ≥ 1), which gives a traversal. Explicitly:
etc.
This can be interpreted as mapping the infinite depth binary tree onto this tree and then applying breadthfirst search: replace the "down" edges connecting a parent node to its second and later children with "right" edges from the first child to the second child, from the second child to the third child, etc. Thus at each step one can either go down (append a (, 1) to the end) or go right (add one to the last number) (except the root, which is extra and can only go down), which shows the correspondence between the infinite binary tree and the above numbering; the sum of the entries (minus one) corresponds to the distance from the root, which agrees with the 2^{n−1} nodes at depth n − 1 in the infinite binary tree (2 corresponds to binary).
By: Wikipedia.org
Edited: 20210618 14:30:13
Source: Wikipedia.org