notebook/notes/set/trees.md

title	TARGET DECK	FILE TAGS	tags
Trees	Obsidian::STEM	set::tree	graph set tree

Overview

A free tree is a connected, acyclic, undirected graphs. If an undirected graph is acyclic but possibly disconnected, it is a forest.

%%ANKI Basic What is a free tree? Back: A connected, acyclic, undirected graph. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is a forest? Back: An acyclic undirected graph. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What additional property must an undirected graph exhibit to be a forest? Back: It must be acyclic. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What additional properties must an undirected graph exhibit to be a free tree? Back: It must be acyclic and connected. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What additional properties must a forest exhibit to be a free tree? Back: It must be connected. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What additional properties must a free tree exhibit to be a forest? Back: N/A. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic If the following isn't a free tree, why not? ! Back: N/A Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic If the following isn't a free tree, why not? ! Back: Because it is disconnected. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic If the following isn't a free tree, why not? ! Back: Because it contains a cycle. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic If the following isn't a forest, why not? ! Back: N/A Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic If the following isn't a forest, why not? ! Back: N/A. It is. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic If the following isn't a forest, why not? ! Back: Because it contains a cycle. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How do free trees pictorially relate to forests? Back: A forest is drawn as one or more free trees. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Rooted Trees

A rooted tree is a free tree in which one vertex is distinguished/blessed as the root. We call vertices of rooted trees nodes.

Let T be a rooted tree with root r. Any node y on the simple path from r to node x is an ancestor of x. Likewise, x is a descendant of y. If the last edge on the path from r to x is \{y, x\}, y is the parent of x and x is a child of y. Nodes with the same parent are called siblings.

A node with no children is an external node or leaf. A node with at least one child is an internal node or nonleaf. The number of children of a node is the degree of said node. The length of the simple path from the root to a node x is the depth of x in T. A level of a tree consists of all nodes at the same depth. The height of a node in a tree is the length of the longest simple path from the node to a leaf.

%%ANKI Basic What is a rooted tree? Back: A free tree in which one of the vertices is distinguished from the others. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is every rooted tree a free tree? Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is every free tree a rooted tree? Back: No. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How many levels exist in a rooted tree of height h? Back: h + 1 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the height of a rooted tree with k levels? Back: k - 1 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which free trees are not considered rooted trees? Back: Those without some vertex identified as the root. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What distinguishes a node from a vertex? Back: A node is a vertex of a rooted tree. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is every vertex a node? Back: No. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is every node a vertex? Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze {Nodes} are to rooted trees whereas {vertices} are to free trees. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which of free trees or rooted trees is a more general concept? Back: Free trees. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for node y to be an ancestor of node x in a rooted tree? Back: The simple path from the root to x contains y. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for node y to be a descendent of node x in a rooted tree? Back: The simple path from the root to y contains x. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze In a rooted tree, if y is an {ancestor} of x, then x is a {descendant} of y. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the ancestors of a rooted tree's root? Back: Just the root itself. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the descendants of a rooted tree's root? Back: Every node in the tree. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the proper ancestors of a rooted tree's root? Back: There are none. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the proper descendants of a rooted tree's root? Back: Every node but the root. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for node y to be a child of node x in a rooted tree? Back: There exists a path from the root to y such that the last edge is \{x, y\}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for node y to be a parent of node x in a rooted tree? Back: There exists a path from the root to x such that the last edge is \{y, x\}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a rooted tree, how does the concept of "ancestor" relate to "parent"? Back: Ancestors include parents, parents of parents, etc. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a rooted tree, how does the concept of "descendants" relate to "child"? Back: Descendants include children, children of children, etc. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a rooted tree, how many ancestors does a node have? Back: At least one (i.e. itself). Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a rooted tree, how many parents does a node have? Back: Zero or one. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a rooted tree, how many descendants does a node have? Back: At least one (i.e. itself). Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a rooted tree, how many children does a node have? Back: Zero or more. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which nodes in a rooted tree has no parent? Back: Just the root. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a rooted tree, what are siblings? Back: Nodes that have the same parent. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a rooted tree, what is an external node? Back: A node with no children. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a rooted tree, what alternative term is used in favor of "external node"? Back: A leaf. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a rooted tree, what is an internal node? Back: A node with at least one child. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a rooted tree, what alternative term is used in favor of "internal node"? Back: A nonleaf. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze {1:External} nodes are to {2:leaf} nodes whereas {2:internal} nodes are to {1:nonleaf} nodes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let T be a rooted tree. What does the degree of a node refer to? Back: The number of children that node has. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let T be a rooted tree. What does the depth of a node refer to? Back: The length of the simple path from the root to the node. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let T be a rooted tree. What does a level refer to? Back: A set of nodes in T that have the same depth. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let T be a rooted tree. What does the height of a node refer to? Back: The length of the longest simple path from said node to a leaf. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the height of a rooted tree in terms of "height"? Back: The height of its root. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the height of a rooted tree in terms of "depth"? Back: The largest depth of any node in the tree. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let T be a rooted tree of height h. Which nodes have height 0? Back: The external nodes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let T be a rooted tree of height h. Which nodes have height h? Back: The root node. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let T be a rooted tree of height h. Which nodes have depth 0? Back: The root. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Let T be a rooted tree of height h. Which nodes have depth h? Back: The external nodes on the longest simple paths from the root to said nodes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the height of this rooted tree? ! Back: 4 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the height of node 4 in the following rooted tree? ! Back: 1 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the depth of node 11 in the following rooted tree? ! Back: 2 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which node has the largest depth in the following rooted tree? ! Back: 9 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which node has the largest height in the following rooted tree? ! Back: 7 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which nodes are on level 3 in the following rooted tree? ! Back: 1, 6, and 5. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which level has the most nodes in the following rooted tree? ! Back: The second level. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which nodes have depth corresponding to this rooted tree's height? ! Back: 9 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which nodes have the most siblings in the following rooted tree? ! Back: 3, 10, and 4 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which nodes are ancestors to 12 in the following rooted tree? ! Back: 12, 3, and 7. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which nodes are descendants of 4 in the following rooted tree? ! Back: 4, 11, and 2. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which nodes are parents of 6 in the following rooted tree? ! Back: 8 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which nodes are children of 7 in the following rooted tree? ! Back: 3, 10, and 4 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the internal nodes of the following rooted tree? ! Back: 7, 3, 4, 12, 8, and 5. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What are the external nodes of the following rooted tree? ! Back: 10, 11, 2, 1, 6, and 9. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What level does node 6 reside on in the following rooted tree? ! Back: 3 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Ordered Trees

An ordered tree is a rooted tree in which the children of each node are ordered.

%%ANKI Basic What is an ordered tree? Back: A rooted tree in which the children of each node are ordered. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which of ordered trees or rooted trees is the more general concept? Back: Rooted trees. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which of free trees or ordered trees is the more general concept? Back: Free trees. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is every rooted tree an ordered tree? Back: No. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is every ordered tree a rooted tree? Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic The following two trees are equivalent when considered as what (most specific) kind of trees? ! Back: Rooted trees. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic The following two trees are different when considered as what (most general) kind of trees? ! Back: Ordered trees. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Considered as rooted trees, are the following trees the same? ! Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Considered as ordered trees, are the following trees the same? ! Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Considered as positional trees, are the following trees the same? ! Back: No. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Considered as binary trees, are the following trees the same? ! Back: No. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Why are these two binary trees not the same? ! Back: 5 is a left child in the first tree but a right child in the second. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What O(n) space representation is commonly used for ordered trees with unbounded branching? Back: A left-child, right-sibling tree representation. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic A node of a left-child, right-sibling tree representation has what three pointers? Back: The parent, left child, and right sibling. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the space usage of a left-child, right-sibling representation? Back: Given n nodes in the tree, O(n). Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What space may be wasted in a k-child representation of a k-ary tree? Back: Some children may be absent. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What space advantage does a left-child, right-sibling representation have over a k-child representation? Back: Absent children are not stored in the former. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How is a struct of a k-child tree representation written? Back:

struct Node {
  struct Node *parent;
  struct Node *children[k];
};

Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: c17

END%%

%%ANKI Basic What tree representation corresponds to the following struct?

struct Node {
  struct Node *parent;
  struct Node *children[k];
};

Back: A k-child representation. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: c17

END%%

%%ANKI Basic How is a struct of a left-child, right-sibling tree representation written? Back:

struct Node {
  struct Node *parent;
  struct Node *left;
  struct Node *next;
};

Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: c17

END%%

%%ANKI Basic What tree representation corresponds to the following struct?

struct Node {
  struct Node *parent;
  struct Node *left;
  struct Node *next;
};

Back: A left-child, right-sibling representation. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: c17

END%%

%%ANKI Basic What is an LCRS tree representation? Back: A left-child, right-sibling representation. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic The following is a portion of what kind of tree representation? ! Back: A left-child, right-sibling representation. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic The following is a portion of what kind of tree representation? ! Back: A k-child (binary) representation. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Positional Trees

A positional tree is a rooted tree in which each child is labeled with a specific positive integer. A k-ary tree is a positional tree with at most k children/labels. A binary-tree is a 2-ary tree.

A k-ary tree is full if every node has degree 0 or k. A k-ary tree is perfect if all leaves have the same depth and all internal nodes have degree k. A k-ary tree is complete if the last level is not filled but all leaves have the same depth and are leftmost arranged.

%%ANKI Basic Why aren't terms "complete/perfect" and "nearly complete/complete" quite synonymous? Back: In the former, "perfect" trees are a subset of "complete" trees. Reference: “Binary Tree,” in Wikipedia, March 13, 2024, https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees.

END%%

%%ANKI Basic What distinguishes a positional tree from a k-ary tree? Back: A k-ary tree cannot have child with label > k. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is a k-ary tree a positional tree? Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is a positional tree a k-ary tree? Back: Not necessarily. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What distinguishes positional trees from ordered trees? Back: Children of the former are labeled with a distinct positive integer. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is the notion of absent children a concept in ordered trees? Back: No. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is the notion of absent children a concept in positional trees? Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is the notion of absent children a concept in k-ary trees? Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is a positional tree? Back: A rooted tree in which each child is labeled with a distinct positive integer. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is a k-ary tree? Back: A positional tree with labels greater than k missing. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which of positional trees or k-ary trees are more general? Back: The positional tree. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Which of positional trees or ordered trees are more general? Back: N/A. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Is the concept of fullness related to positional trees or k-ary trees? Back: k-ary trees. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%%

%%ANKI Basic Is the concept of perfectness related to positional trees or k-ary trees? Back: k-ary trees. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is the concept of completeness related to positional trees or k-ary trees? Back: k-ary trees. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for a k-ary tree to be full? Back: Each node has 0 or k children. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What degrees are permitted in a full k-ary tree? Back: 0 and k. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What degrees are permitted in a perfect k-ary tree? Back: 0 and k. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for a k-ary tree to be perfect? Back: All leaves have the same depth and all internal nodes have degree k. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the degree of an internal node in a perfect k-ary tree? Back: k Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the degree of an external node in a perfect k-ary tree? Back: 0 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What recursive definition describes the number of nodes in each level of a perfect k-ary tree? Back: a_n = k \cdot a_{n-1} with a_0 = 1 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: algebra::sequence

END%%

%%ANKI Basic How many nodes are in a perfect k-ary tree of height h? Back: \frac{1 - k^{h+1}}{1 - k} Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: algebra::sequence

END%%

%%ANKI Basic How many internal nodes are in a perfect k-ary tree of height h? Back: \frac{1 - k^h}{1 - k} Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: algebra::sequence

END%%

%%ANKI Basic How many external nodes are in a perfect k-ary tree of height h? Back: k^h Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: algebra::sequence

END%%

%%ANKI Basic How many nodes are on level d of a perfect k-ary tree? Back: k^d Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: algebra::sequence

END%%

%%ANKI Basic What kind of sequence describes the number of nodes in a perfect k-ary tree? Back: A geometric sequence. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: algebra::sequence

END%%

%%ANKI Basic What is the common ratio of the geometric sequence used to count nodes of a perfect k-ary tree? Back: k Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: algebra::sequence

END%%

%%ANKI Basic What does it mean for a k-ary tree to be complete? Back: All levels, except maybe the last, are filled. All leaves have the same depth and are leftmost arranged. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How is the minimum number of nodes in a complete k-ary tree of height h calculated in terms of perfect k-ary trees? Back: As "the number of nodes in a perfect k-ary tree of height h - 1" plus 1. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the maximum number of nodes in a complete binary tree of height h? Back: 2^{h+1} - 1 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How is the maximum number of nodes in a complete k-ary tree of height h calculated in terms of perfect k-ary trees? Back: As "the number of nodes in a perfect k-ary tree of height h". Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How many internal nodes are in a complete k-ary tree of n nodes? Back: \lceil (n - 1) / k \rceil Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What value of k is used in the following description of a complete k-ary tree?

\begin{array}{c|c|c}
n & \text{external} & \text{internal} \\
\hline
1 & 1 & 0 \\
2 & 1 & 1 \\
3 & 2 & 1 \\
4 & 3 & 1 \\
5 & 4 & 1 \\
6 & 4 & 2 \\
7 & 5 & 2 \\
8 & 6 & 2
\end{array}$$
Back: $4$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714349367637-->
END%%

%%ANKI
Basic
What value of $k$ is used in the following description of a complete $k$-ary tree?
$$\begin{array}{c|c|c}
n & \text{external} & \text{internal} \\
\hline
1 & 1 & 0 \\
2 & 1 & 1 \\
3 & 2 & 1 \\
4 & 2 & 2 \\
5 & 3 & 2 \\
6 & 3 & 3 \\
7 & 4 & 3 \\
8 & 4 & 4
\end{array}$$
Back: $2$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714349367640-->
END%%

%%ANKI
Basic
When does the number of external nodes increment in a growing $k$-ary tree?
Back: When the next node added already has a sibling.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714349367644-->
END%%

%%ANKI
Basic
When does the number of external nodes remain static in a growing $k$-ary tree?
Back: When the next node added has no sibling.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714349367647-->
END%%

%%ANKI
Basic
When does the number of internal nodes increment in a growing $k$-ary tree?
Back: When the next node added has no sibling.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714349367651-->
END%%

%%ANKI
Basic
When does the number of internal nodes remain static in a growing $k$-ary tree?
Back: When the next node added already has a sibling.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714349367655-->
END%%

## Bibliography

* Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).