Question

Let \(T\) be an ordered tree with more than one node. Is it possible that the preorder traversal of \(T\) visits the nodes in the same order as the postorder traversal of \(T ?\) If so, give an example; otherwise, argue why this cannot occur. Likewise, is it possible that the preorder traversal of \(T\) visits the nodes in the reverse order of the postorder traversal of \(T ?\) If so, give an example; otherwise, argue why this cannot occur.

Short answer

It is impossible for preorder and postorder traversals of a tree with more than one node to be either the same or reverse of each other.

Step-by-step solution

- Understand Traversal Orders

Preorder traversal visits the root node first, then recursively visits the left subtree, and finally visits the right subtree. Postorder traversal first recursively visits the left subtree, then the right subtree, and finally visits the root node.

- Analyze the Possibility of Same Order

For preorder and postorder traversals to be the same, the root must be both the first and last node visited. Consider the properties of preorder and postorder: in preorder, the root is always first, and in postorder, the root is always last.

- Conclusion for Same Order

It's impossible for the preorder and postorder traversals of a tree with more than one node to be the same. In any tree with more than one node, the root can't simultaneously be at the start and the end of the sequence.

- Analyze the Possibility of Reverse Order

For preorder traversal to be the reverse of postorder traversal, each subtree visited must fulfill this condition individually. Specifically, the root must be visited after all its subtrees in preorder and before all its subtrees in postorder.

- Find a Counterexample

Considering the nature of traversal orders, it's impossible for the preorder traversal of a tree to be the reverse order of the postorder traversal for any tree with more than one node. The root would need to be both last and first in the sequences.

- Conclusion for Reverse Order

Similarly, it's impossible for the preorder traversal to be the reverse of the postorder traversal.

Key concepts

Preorder Traversal

Preorder traversal is a method of visiting all the nodes in a tree data structure.
In this method, you visit the root node first.
Next, you recursively visit the left subtree, and finally, the right subtree.
This sequence ensures that the root is always the first node to be visited.
A key thing to remember is the order:
root, left subtree, and then right subtree.
For example, let's consider a simple ordered tree:

• A (root)
• B (left child of A)
• C (right child of A)

In preorder traversal, we would visit A first, then B, and finally C.
Thus, the order would be A, B, C.
This traversal method is particularly useful for creating a copy of the tree.
It can also be useful for prefix notation in expressions.

Postorder Traversal

Postorder traversal differs quite significantly from preorder traversal.
In this method, you begin by recursively visiting the left subtree.
Then you visit the right subtree, and finally, the root node.
The crucial aspect here is that the root is always the last node visited.
So, the order of visitation is:
left subtree, right subtree, and then the root.
Let's take the same tree as before:

• A (root)
• B (left child of A)
• C (right child of A)

By using postorder traversal, we would visit B first, then C, and finally, A.
Thus, the order would be B, C, and A.
Postorder traversal is very useful for deleting nodes and freeing memory.
It's also commonly used for evaluating postfix expressions and syntax trees in compilers.

Ordered Tree

An ordered tree is a tree in which the order of the children is significant.
This means if you change the order of the children nodes, you will get a different tree.
Often, this ordering is utilized in various applications, such as compilers and databases.
In an ordered tree, traversals like preorder or postorder can give different sequences when the order of children changes.
The importance of ordered trees lies in their structured nature.
This makes them ideal for situations where the order of processing nodes matters.
For example, an ordered tree can be used to represent mathematical expressions.
In this context, the order will affect the result of the computation.
Consider another simple tree:

• A (root)
• D (left child of A)
• E (right child of A)

If the sequence is changed to:

• A (root)
• E (left child of A)
• D (right child of A)

Then, the preorder traversal will be different while the tree structure remains consistent.