What Are Tree Traversals (Inorder, Preorder, and Postorder) – With Examples

Tree data structures are a critical component of computer science and are used to efficiently organize hierarchical data. Performing operations on tree nodes requires traversing the tree to access each element, which is done through specialized algorithms known as tree traversals. Mastering common traversal techniques like inorder, preorder, and postorder is key for any developer working with tree-based data.

In this comprehensive guide, we will explain what tree traversals are, provide clear examples of the main traversal methods, discuss their applications, and answer common questions around this pivotal concept.

Understanding Tree Structures

Before diving into specifics on traversals, it‘s important to level-set on some key properties of trees:

• Trees consist of a root node plus additional nodes connected in a hierarchy via edges
• The root node is the single node at the top of the tree structure
• All other nodes stem from the root forming branches and sub-branches
• Leaf nodes have no children while internal nodes have one or more child nodes
• Node relationships include parent, child, sibling

Trees provide an intuitive way to represent hierarchical data like organizational charts, folders on a hard drive, DOM elements on a web page, etc. The connections between elements are visualized for rapid comprehension.

In computer science, the most common tree structure is the binary tree where each parent node has a maximum of two children. Binary trees strike an optimal balance between flexibility and efficiency.

Why Tree Traversals Matter

Being able to efficiently traverse all nodes in a tree is critical for many common operations:

• Searching – locate a specific node like finding a folder on a hard drive
• Sorting – arrange nodes in a certain order like alphabetizing files
• Insertion/Deletion – add and remove elements from the tree
• Copying/Cloning – duplicate the entire tree structure

Rather than manually visiting each node, specialized traversal algorithms handle the node-by-node traversal programmatically. The order in which nodes are accessed differs among techniques, producing different outputs.

Depth-First vs Breath-First Traversal

At the highest level, there are two categories of tree traversal algorithms:

Depth-First – nodes on same branch fully traversed before moving laterally

Breadth-First – nodes closest to root visited before moving down hierarchy

Within depth-first, there are three main methods: preorder, inorder, and postorder differing in when the root node is accessed relative to the children. Breadth-first employs one strategy working top-down.

Let‘s explore each approach more closely.

Preorder Tree Traversal

The preorder traversal method accessed the root node first, then recursively visits the left subtree, then right subtree. In code, it follows this order:

1. Process root node
2. Traverse left subtree
3. Traverse right subtree

When visualized on a sample tree, this produces the following order:

Here is the standard algorithm defining a preorder traversal:

preorder(tree)

  1. Visit root node

  2. If left subtree exists, traverse left subtree   
       preorder(left-subtree)

  3. If right subtree exists, traverse right subtree 
       preorder(right-subtree) 

The key advantage of preorder traversal is efficiency in copying or cloning a tree since the root node is accessed first before descending into branches. It also provides quick access to root and left nodes which may be prioritized.

Inorder Tree Traversal

Inorder traversal first visits the left child node, then processes the root node, and finally visits the right child node, following this order:

1. Traverse left subtree
2. Process root node
3. Traverse right subtree

When performed on our sample binary tree, an inorder traversal works through the nodes in the following sequence:

The inorder algorithm is defined as:

inorder(tree)

  1. If left subtree exists, traverse left subtree  
       inorder(left-subtree)

  2. Visit root node

  3. If right subtree exists, traverse right subtree   
       inorder(right-subtree)  

An important characteristic of inorder traversal is the sorted output it generates. All left children are accessed before parents before right children. This runs through a binary search tree producing ascending sequential values perfect for sorting duties.

Postorder Tree Traversal

The final depth-first traversal method is called postorder. It visits nodes in this order:

1. Traverse left subtree
2. Traverse right subtree
3. Process root node

Our sample binary tree undergoes postorder traversal in the following manner:

Here is the standard postorder algorithm:

postorder(tree)

   1. If left subtree exists, traverse left subtree   
         postorder(left-subtree)

   2. If right subtree exists, traverse right subtree
        postorder(right-subtree)  

   3. Visit root node

The postorder method is extremely useful for deleting or freeing a tree from memory because branches can be removed before the root node. All child nodes are accessed first before finishing at the root.

Breadth-First Traversal

The only form of breadth-first traversal is achieved using a queue to visit nodes level-by-level starting closest to the root. It follows this order:

1. Visit root node and enqueue
2. Dequeue node and visit all children
3. Enqueue any children
4. Repeat until queue empty

Which completes this sample tree in the following sequence:

The algorithm drives from a queue:

breadthFirst(root)

  1. Enqueue root node
  2. While queue not empty
         - Dequeue node
         - Visit node 
         - If node has left child
               - Enqueue left child
         - If node has right child    
               - Enqueue right child

The breadth-first approach prioritizes nodes higher up versus drilling down immediately. Useful for analyzing relationships and hierarchies.

Comparing Traversal Methods

This summarizes how each algorithm uniquely traverses a tree. Preorder prioritizes root access, inorder generates sequential output, postorder finishes at the root, and breadth-first goes level by level.

Traversal Algorithms in Practice

Let‘s apply all four traversal methods on an example family tree to better understand how they work in practice:

• Preorder – Grandpa Joe, Uncle Bob, Aunt Jane, Dad, Aunt Lisa, Mom, Me
• Inorder – Aunt Jane, Uncle Bob, Grandpa Joe, Dad, Mom, Aunt Lisa, Me
• Postorder – Aunt Jane, Uncle Bob, Dad, Aunt Lisa, Mom, Grandpa Joe, Me
• Breadth-First – Grandpa Joe, Uncle Bob, Aunt Jane, Dad, Mom, Aunt Lisa, Me

This shows how unique node orderings are achieved mirroring the logic defined in each algorithm earlier.

Why Mastering Tree Traversals Matters

Whether implementing a system to recommend online content to users, serve up search results, process e-commerce orders, route network packets, or store massive datasets, trees enable organizing complex hierarchical information for rapid digestion by humans and computers alike.

Fluency with inorder, preorder, postorder, and breadth-first traversals unlocks the ability to effectively construct, analyze, manipulate, and interpret tree-based data structures for a variety of practical applications. Take time to practice applying the various algorithms by hand to small sample trees to cement your competency with this vital skill.

Core Takeaways on Tree Traversals

• Traversals allow access to all nodes within trees via specialized algorithms
• Depth-first traversals work vertically accessing branches before siblings
• Common depth-first methods include preorder, inorder, and postorder
• Breadth-first traversal visits levels horizontally starting at root
• Each algorithm offers unique node ordering and use cases
• Mastering traversals key for managing real-world tree data sets

Frequently Asked Questions

What are the parts of a tree data structure?

Trees contain a root node, leaf nodes, child nodes, parent nodes, edges, depth, height, subtrees, and siblings.

When would I use postorder traversal versus breadth-first?

Use postorder when you want to free memory from the bottom up. Use breadth-first for analyzing hierarchical data level-by-level starting from the top.

Is inorder traversal the same as sorted order?

Yes, inorder traversal generates output as if the tree was sorted depth-first from left to right branches.

What trees cannot be traversed?

Cyclic trees with looping branches cannot be traversed because algorithms will loop infinitely. Trees must have a defined root with directed edges.

How are binary search trees related to inorder traversal?

Performing inorder traversal on a binary search tree will visit nodes in numeric/alphabetic order, which defines this structure.