Traversal techniques

TREE TRAVERSAL TECHNIQUES:

Pre-order: F, B, A, D, C, E, G, I, H

In-order: A, B, C, D, E, F, G, H, I

Post-order: A, C, E, D, B, H, I, G, F

Traversing a tree involves iterating (looping) 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 self-referential (recursively defined) data structure, traversal can naturally be described by Recursion or  more subtly Co-recusrion, in which case the deferred nodes are stored implicitly – in the case of recursion, in the Call Stack.

Depth-first traversal is easily implemented via a stack, including recursively (via the call stack), while breadth-first traversal is easily implemented via a queue, including corecursively.

Beyond these basic traversals, various more complex or hybrid schemes are possible, such as Depth-limited Searches such as iterative depth-first-search.

There are three types of depth-first traversal: pre-order in-order, and post-order.For a binary tree, they are defined as operations recursively at each node, starting with the root node as follows:Pre-order

1. Visit the root.

1. Traverse the left subtree.

1. Traverse the right subtree.

In-order (symmetric)

1. Traverse the left subtree.

1. Visit the root.

1. Traverse the right subtree.

Post-order

1. Traverse the left subtree.

1. Traverse the right subtree.

1. Visit the root.

The trace of a traversal is called a sequentialisation of the tree. The traversal trace is a list of each visited root node. No one sequentialisation according to pre-, in- or post-order describes the underlying tree uniquely. Given a tree with distinct elements, either pre-order or post-order paired with in-order is sufficient to describe the tree uniquely. However, pre-order with post-order leaves some ambiguity in the tree structure.
Generic treeTo traverse any tree in depth-first order, perform the following operations recursively at each node:

1. Perform pre-order operation

1. For each i (with i = 1 to n − 1) do:

1. Visit i-th, if present

1. Perform in-order operation

1. Visit n-th (last) child, if present

1. Perform post-order operation

where n is the number of child nodes. Depending on the problem at hand, the pre-order, in-order or post-order operations may be void, or you may only want to visit a specific child node, so these operations are optional. Also, in practice more than one of pre-order, in-order and post-order operations may be required. For example, when inserting into a ternary tree, a pre-order operation is performed by comparing items. A post-order operation may be needed afterwards to re-balance the tree.Breadth-first.

Trees can also be traversed in level-order, where we visit every node on a level before going to a lower level. This search is referred to as bfs, as the search tree is broadened as much as possible on each depth before making going to the next depth.Other types

There are also tree traversal algorithms that classify as neither depth-first search nor breadth-first 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.

ImplementationsPre-order traversal while duplicating nodes and edges can make a complete duplicate of a binary tree.

 It can also be used to make a prefix expression (Polish notation) from expression trees: traverse the expression tree pre-orderly.

In-order 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 (hence the name).

Post-order traversal while deleting or freeing nodes and values can delete or free an entire binary tree. It can also generate a  postfix  representation of a binary tree.
Infinite treesDepth-first

Pre-order

preorder(node) if node == null then return visit(node) preorder(node.left) preorder(node.right)

iterativePreorder(node) parentStack = empty stack while (not parentStack.isEmpty() or node ≠ null) if (node ≠ null) visit(node) if (node.right ≠ null) parentStack.push(node.right) node = node.left else node = parentStack.pop()

In-order

inorder(node) if node == null then return inorder(node.left) visit(node) inorder(node.right)

iterativeInorder(node) parentStack = empty stack while (not parentStack.isEmpty() or node ≠ null) if (node ≠ null) parentStack.push(node) node = node.left else node = parentStack.pop() visit(node) node = node.right

Post-order

postorder(node) if node == null then return postorder(node.left) postorder(node.right) visit(node)

iterativePostorder(node) parentStack = empty stack lastnodevisited = null while (not parentStack.isEmpty() or node ≠ null) if (node ≠ null) parentStack.push(node) node = node.left else peeknode = parentStack.peek() if (peeknode.right ≠ null and lastnodevisited ≠ peeknode.right) /* if right child exists AND traversing node from left child, move right */ node = peeknode.right else visit(peeknode) lastnodevisited = parentStack.pop()

All the above implementations require call stackspace proportional to the height of the tree. In a poorly balanced tree, this can be considerable. We can remove the stack requirement by maintaining parent pointers in each node, or by threading the tree (next section).Morris in-order traversal using threadingA binary tree is threaded by making every left child pointer (that would otherwise be null) point to the in-order predecessor of the node (if it exists) and every right child pointer (that would otherwise be null) point to the in-order successor of the node (if it exists).Advantages:

1. Avoids recursion, which uses a call stack and consumes memory and time.

1. The node keeps a record of its parent.

Disadvantages:

1. The tree is more complex.

1. We can make only one traversal at a time.

1. It is more prone to errors when both the children are not present and both values of nodes point to their ancestors.

Morris traversal is an implementation of in-order traversal that uses threading:

1. Create links to the in-order successor

1. Print the data using these links

1. Revert the changes to restore original tree.

Breadth-firstAlso, listed below is pseudocode for a simple queue based level order 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 space-efficient approach for this type of traversal can be implemented using an iterative deepening depth-first search.levelorder(root) q = empty queue q.enqueue(root)while not q.empty do node := q.dequeue() visit(node)if node.left ≠ null then q.enqueue(node.left)if node.right ≠ null then q.enqueue(node.right)ReferencesWhile traversal is usually done for trees with a finite number of nodes (and hence finite depth and finite Branching Factor).

A basic requirement for traversal is to visit every node. For infinite trees, simple algorithms often fail this. For example, given a binary tree of infinite depth, a depth-first traversal will go down one side (by convention the left side) of the tree, never visiting the rest, and indeed if in-order or post-order will never visit any nodes, as it has not reached a leaf (and in fact never will). 

By contrast, a breadth-first (level-order) 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 node has infinitely many children, and each of these children has two children, a depth-first traversal 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 post-order, in which case it never reaches the root). 

By contrast, a breadth-first traversal will never reach the grandchildren, as it seeks to exhaust the children first.
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("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 node  the children of the root node  the grandchildren  and so on. The nodes are thus in a 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 Lexographic-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 of a given natural number, specifically 2^n-1compositions of n = 1;), which gives a traversal. Explicitly:0: ()1: (1)2: (1,1) (2)3: (1,1,1) (1,2) (2,1) (3)4: (1,1,1,1) (1,1,2) (1,2,1) (1,3) (2,1,1) (2,2) (3,1) (4).

This can be interpreted as mapping the infinite depth binary tree onto this tree and then applying breadth-first traversal: replace the "down" edges connecting a parent node to its second and later children with "right" edges from the 1st child to the 2nd child, 2nd child to third child, etc. Thus at each step one can either go down (append a (,1) to the end) or go right (add 1 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 1) 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).

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