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Chapter 9: Problem 20

What can you say about a vertex in a rooted tree that has no descendants?

Short Answer

Expert verified
A vertex in a rooted tree that has no descendants is called a leaf or terminal node.

Step by step solution

01

Understanding Rooted Tree

A rooted tree is a tree in which one vertex has been designated as the root and the direction of the edges are directed away or towards the root. The vertices are often referred to as nodes, and these nodes can have zero or more descendant nodes originating from them.
02

Understanding Vertex with No Descendants

In a rooted tree, any node that has no descendants is called a 'leaf' or 'terminal' node. This means that it does not have any edges emanating from it to other nodes.
03

Conclusion

So, a vertex in a rooted tree that has no descendants is a leaf node. It represents the 'end' of a path within the tree.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex
In the realm of data structures and trees, a "vertex" is a fundamental concept. It's essentially a point in the tree where lines, known as edges, meet. If you picture a tree, each point where the lines meet or branch is called a vertex. Everything in the tree begins or ends at these vertices.
  • Vertices are the building blocks of trees and graphs.
  • Every vertex holds a value or piece of data.
  • Vertices are connected to each other through edges, forming the tree structure.

In any tree structure, the relationship between different vertices dictates the organization and hierarchy of the data contained within the tree.
Descendants
In a rooted tree, the term "descendants" refers to all the other nodes that branch off from a given vertex. This includes all nodes in the paths that extend downward from a vertex.
  • A node's descendants include its children, grandchildren, and so on.
  • Descendants show the path from one vertex to another, and the relationships between them.
  • If a vertex has no descendants, it is known as a leaf node, indicating the end of that path in the tree.

Understanding the concept of descendants helps in navigating through the tree and locating specific nodes within the hierarchy.
Leaf Node
A "leaf node" is a unique vertex in a rooted tree that has no descendants. This means it does not lead to any other vertices. In terms of a tree, a leaf node is like the leaf on a branch that does not branch out further.
  • Leaf nodes do not have any children nodes.
  • They represent the endpoint of a path through the tree.
  • Leaf nodes are significant because they can indicate complete data pathways or terminal operations in various applications.

The simplicity of leaf nodes often makes them important in algorithms that traverse trees, as they denote the conclusion of a particular traversal path.
Tree Structure
A "tree structure" is an arrangement of vertices and edges that resemble a hierarchy or branching pattern, much like a natural tree. At the top of a tree structure is a sole root vertex, with branches leading away to connect to other vertices.
  • A tree has a single root from which all nodes descend.
  • Each node may have zero or more child nodes.
  • The hierarchical structure is used for modeling relationships and data organization.

Tree structures are used in numerous fields of computer science, including network design, database management, and decision-making processes due to their ability to represent complex hierarchical data simply and effectively.

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Most popular questions from this chapter

Draw all nonisomorphic binary trees having two vertices.

Draw all nonisomorphic rooted trees having five vertices.

Show that a graph \(G\) with \(n\) vertices and fewer than \(n-1\) edges is not connected.

Draw all nonisomorphic free trees having four vertices.

Refer to the following situation. Suppose that we have stamps of various denominations and that we want to choose the minimum number of stamps to make a given amount of postage. Consider a greedy algorithm that selects stamps by choosing as many of the largest denomination as possible, then as many of the second-largest denomination as possible, and so on. Suppose that the available denominations are \(1=a_{1}<\) \(a_{2}
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