Question

Let \(T\) be an \(n\) -node improper binary tree (that is, each internal node has one or two children). Describe how to represent \(T\) by means of a proper binary tree \(T^{\prime}\) with \(O(n)\) nodes.

Short answer

Insert dummy nodes to replace missing children in an improper binary tree to create a proper binary tree with O(n) nodes.

Step-by-step solution

- Understand improper binary tree

An improper binary tree is one where some internal nodes can have only one child. In this exercise, we need to convert it into a proper binary tree where each internal node has exactly two children.

- Insert dummy nodes

To change the improper tree into a proper binary tree, add a dummy node wherever an internal node has only one child. This ensures all internal nodes have exactly two children.

- Build the proper binary tree

Start from the root. Traverse the tree and for each node with one child, insert a dummy node as the missing second child. Continue this recursively for all such nodes.

- Ensure O(n) nodes

By adding only one dummy node for each internal node with one child, the total number of nodes in the resulting tree, including added dummy nodes, remains O(n). This is because each original node triggers the addition of at most one dummy node.

- Represent with dummy nodes

The final proper binary tree will have all nodes (including dummy nodes) arranged such that each internal node has two children.

Key concepts

Improper Binary Tree

An improper binary tree is a binary tree where internal nodes may have only one child instead of the typical two.
In an improper binary tree, nodes can have the following configurations:

• No children (leaf node).
• One child (improper for binary trees).
• Two children (standard for binary trees).

Improper binary trees arise in various circumstances. They require transformation to a proper format for certain algorithms or data structure requirements.
Understanding improper binary trees is essential for this transformation process.

Proper Binary Tree

A proper binary tree is a binary tree where every internal node has exactly two children.
This consistent structure simplifies many tree operations and algorithms because each node conforms to the expected binary format.
To transform an improper binary tree into a proper binary tree, you must ensure every internal node has two children. This transformation relies heavily on the concept of dummy nodes, which we will explore next.

Dummy Nodes

Dummy nodes are placeholder nodes used to convert an improper binary tree into a proper binary tree.
Here's how they work:

• Identify internal nodes with only one child.
• Insert a dummy node as the missing second child for each of these nodes.

For example, if an internal node has only a left child, add a dummy node as its right child. Conversely, if it has only a right child, add a dummy node as its left child.
This ensures every internal node conforms to the proper binary tree structure. By carefully placing dummy nodes, we can convert any improper binary tree (with nodes having one or two children) into a proper binary tree with an increased number of nodes that is still manageable within the constraint of O(n).