Question

Describe a generalization of the Euler tour traversal of trees such that each internal node has three children. Describe how you could use this traversal to compute the height of each node in such a tree.

Short answer

Generalize Euler Tour by visiting three children in sequence. Track and record depth at each visit to compute heights.

Step-by-step solution

Understand Euler Tour Traversal

Euler Tour Traversal involves visiting every edge of the tree exactly twice. For a binary tree, this means visiting the root, then its left subtree, followed by its right subtree, and coming back to the root.

Generalize Euler Tour for Ternary Tree

In a ternary tree, each internal node has three children. Modify the Euler Tour by visiting each child from left to right. The sequence would be: visit node, visit left child subtree, visit middle child subtree, visit right child subtree, and then return to the node.

Initiate Heights Calculation

To compute the heights during the traversal, track the current depth. Start with height 0 at the root, incrementing by 1 when moving down a level and decrementing by 1 when moving back up.

Visit and Record Heights

Each time an internal node is visited, record the current depth as the height of that node. Traverse to each of its children and repeat this process, ensuring each node's height is recorded during its initial visit.

Implement Height Calculation

Implement a recursive function to perform the generalized Euler Tour. Pass the current depth as an argument. When a node is first visited, record the height as the current depth. Increment the depth before moving to children, and decrement it after returning.

Key concepts

Tree Traversal Algorithms

Tree traversal algorithms are methods to visit all the nodes in a tree data structure in a systematic manner. They are crucial for many tree-related operations. The three main types of tree traversal are:

• Preorder: Visit the root node first, then recursively visit left subtrees and right subtrees.
• Inorder: Recursively visit the left subtree first, then the root node, and finally the right subtree.
• Postorder: Recursively visit the left subtree, then the right subtree, and finally the root node.

The Euler Tour Traversal is another fascinating method. It involves visiting every edge of the tree exactly twice. This traversal method is invaluable for various operations, such as allocating heights to nodes and spanning across other complex tree tasks. Understanding these algorithms is fundamental for students to effectively navigate and manipulate tree structures.

Computing Node Height

The height of a node in a tree is the number of edges on the longest path from that node to a leaf. Computing the height is an integral part of tree-based algorithms. To calculate the height:

• Start at the root node and set its height as 0.
• For each move down to a child node, increment the height by 1.
• If you move back to a parent node from its child, decrement the height by 1.

This process can be encapsulated in a recursive function. The function should track the current depth and update the height accordingly on visiting each node. This ensures that the height is computed correctly during the traversal.

Generalization of Euler Tour

The Euler Tour Traversal can be generalized to handle trees with more than two children, such as ternary trees, where each internal node has three children. For a ternary tree:

• First, visit the node itself.
• Next, recursively visit the subtree of the left child.
• Then, move to the subtree of the middle child.
• Finally, visit the subtree of the right child before returning to the parent node.

This traversal method still adheres to the principle of visiting each edge twice. During this generalized traversal, the height of each node can be calculated by keeping track of the current depth in the tree, much like done in binary trees. Implementing this generalization can deepen a student's understanding of tree traversal, height computation, and recursive function strategies.