notebook/notes/set/trees.md

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

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 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%%

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 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 to 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%%

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 kind of trees? ! Back: Rooted/free 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 kind of trees? ! Back: Ordered trees. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

A binary tree T is a structure defined on a finite set of nodes that either

• contains no nodes, or
• is composed of three disjoint sets of nodes: a root node, a left subtree, and a right subtree.

%%ANKI Basic What is the base case used in the recursive definition of a binary tree? Back: The empty set. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What recurrence is used in the recursive definition of a binary tree? Back: A binary tree is composed of a root node, a left subtree, and a right subtree. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How should the nil constructor of an inductive binary tree, say Tree, be defined? Back:

| constructor : Tree α

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

END%%

%%ANKI Basic How should the non-nil constructor of an inductive binary tree, say Tree, be defined? Back:

| constructor : α → Tree α → Tree α → Tree α

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

END%%

%%ANKI Basic In the following binary tree type, what name is given to the first argument of node?

inductive Tree α where
| nil : Tree α
| node : α → Tree α → Tree α → Tree α

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

END%%

%%ANKI Basic In the following binary tree type, what name is given to the second argument of node?

inductive Tree α where
| nil : Tree α
| node : α → Tree α → Tree α → Tree α

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

END%%

%%ANKI Basic In the following binary tree type, what name is given to the third argument of node?

inductive Tree α where
| nil : Tree α
| node : α → Tree α → Tree α → Tree α

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

END%%

%%ANKI Basic Given the following binary tree implementation, how do you construct an empty tree?

inductive Tree α where
| nil : Tree α
| node : α → Tree α → Tree α → Tree α

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

END%%

%%ANKI Basic Given the following binary tree implementation, how do you construct a tree with root a, left child b, and right child c?

inductive Tree α where
| nil : Tree α
| node : α → Tree α → Tree α → Tree α

Back: node 'a' (node 'b' nil nil) (node 'c' nil nil) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: lean

END%%

%%ANKI Basic Why isn't a binary tree considered an ordered tree? Back: A left child is distinct from a right child, even if the child is the same in both cases. 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 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%%

Bibliography

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