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---
title: Trees
TARGET DECK: Obsidian::STEM
FILE TAGS: set::graph
tags:
- graph
- set
---
## Overview
A **free tree** is a connected, acyclic, undirected [[graphs|graph]]. 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).
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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).
<!-- ID: 1711136844903 -->
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).
<!-- ID: 1711136844906 -->
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).
<!-- ID: 1711136844909 -->
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).
<!-- ID: 1711136844912 -->
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).
<!-- ID: 1711136844915 -->
END%%
%%ANKI
Basic
If the following isn't a free tree, why not?
![[free-tree.png]]
Back: N/A
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136844918 -->
END%%
%%ANKI
Basic
If the following isn't a free tree, why not?
![[forest.png]]
Back: Because it is disconnected.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136844922 -->
END%%
%%ANKI
Basic
If the following isn't a free tree, why not?
![[cyclic-undirected.png]]
Back: Because it contains a cycle.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136844926 -->
END%%
%%ANKI
Basic
If the following isn't a forest, why not?
![[free-tree.png]]
Back: N/A
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136844930 -->
END%%
%%ANKI
Basic
If the following isn't a forest, why not?
![[forest.png]]
Back: N/A
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136844934 -->
END%%
%%ANKI
Basic
If the following isn't a forest, why not?
![[cyclic-undirected.png]]
Back: Because it contains a cycle.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136844939 -->
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).
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END%%
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## Rooted Trees
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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).
<!-- ID: 1711136844947 -->
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).
<!-- ID: 1711136844951 -->
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).
<!-- ID: 1711136844955 -->
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).
<!-- ID: 1711136844958 -->
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).
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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).
<!-- ID: 1711136844966 -->
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).
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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).
<!-- ID: 1711136844973 -->
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).
<!-- ID: 1711136844976 -->
END%%
%%ANKI
Basic
What does it mean for node $y$ to be an ancestor of node $x$ in a rooted tree?
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Back: The simple path from the root to $x$ contains $y$.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136844980 -->
END%%
%%ANKI
Basic
What does it mean for node $y$ to be a descendent of node $x$ in a rooted tree?
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Back: The simple path from the root to $y$ contains $x$.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136844983 -->
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).
<!-- ID: 1711136844986 -->
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).
<!-- ID: 1711136844989 -->
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).
<!-- ID: 1711136844993 -->
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).
<!-- ID: 1711136844996 -->
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).
<!-- ID: 1711136845000 -->
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).
<!-- ID: 1711136845004 -->
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).
<!-- ID: 1711136845009 -->
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).
<!-- ID: 1711136845015 -->
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).
<!-- ID: 1711136845020 -->
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).
<!-- ID: 1711136845026 -->
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).
<!-- ID: 1711136845031 -->
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).
<!-- ID: 1711136845037 -->
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).
<!-- ID: 1711136845044 -->
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).
<!-- ID: 1711136845051 -->
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).
<!-- ID: 1711136845057 -->
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).
<!-- ID: 1711136845063 -->
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).
<!-- ID: 1711136845072 -->
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).
<!-- ID: 1711136845079 -->
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).
<!-- ID: 1711136845087 -->
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).
<!-- ID: 1711136845093 -->
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).
<!-- ID: 1711136845101 -->
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).
<!-- ID: 1711136845107 -->
END%%
%%ANKI
Basic
Let $T$ be a rooted tree. What does a level refer to?
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Back: A set of nodes in $T$ that have the same depth.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136845114 -->
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).
<!-- ID: 1711136845119 -->
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).
<!-- ID: 1711136845124 -->
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).
<!-- ID: 1711136845131 -->
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).
<!-- ID: 1711136845137 -->
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).
<!-- ID: 1711136845141 -->
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).
<!-- ID: 1711136845145 -->
END%%
%%ANKI
Basic
Let $T$ be a rooted tree of height $h$. Which nodes have depth $h$?
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Back: The external nodes on the longest simple paths from the root to said nodes.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136845150 -->
END%%
%%ANKI
Basic
What is the height of this rooted tree?
![[rooted-tree.png]]
Back: $4$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136845156 -->
END%%
%%ANKI
Basic
What is the height of node $4$ in the following rooted tree?
![[rooted-tree.png]]
Back: $1$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136845164 -->
END%%
%%ANKI
Basic
What is the depth of node $11$ in the following rooted tree?
![[rooted-tree.png]]
Back: $2$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136845172 -->
END%%
%%ANKI
Basic
Which node has the largest depth in the following rooted tree?
![[rooted-tree.png]]
Back: $9$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136845178 -->
END%%
%%ANKI
Basic
Which node has the largest height in the following rooted tree?
![[rooted-tree.png]]
Back: $7$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136845184 -->
END%%
%%ANKI
Basic
Which nodes are on level $3$ in the following rooted tree?
![[rooted-tree.png]]
Back: $1$, $6$, and $5$.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136845191 -->
END%%
%%ANKI
Basic
Which level has the most nodes in the following rooted tree?
![[rooted-tree.png]]
Back: The second level.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136845198 -->
END%%
%%ANKI
Basic
Which nodes have depth corresponding to this rooted tree's height?
![[rooted-tree.png]]
Back: $9$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136845205 -->
END%%
%%ANKI
Basic
Which nodes have the most siblings in the following rooted tree?
![[rooted-tree.png]]
Back: $3$, $10$, and $4$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136845210 -->
END%%
%%ANKI
Basic
Which nodes are ancestors to $12$ in the following rooted tree?
![[rooted-tree.png]]
Back: $12$, $3$, and $7$.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136845214 -->
END%%
%%ANKI
Basic
Which nodes are descendants to $4$ in the following rooted tree?
![[rooted-tree.png]]
Back: $4$, $11$, and $2$.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136845219 -->
END%%
%%ANKI
Basic
Which nodes are parents of $6$ in the following rooted tree?
![[rooted-tree.png]]
Back: $8$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136845223 -->
END%%
%%ANKI
Basic
Which nodes are children of $7$ in the following rooted tree?
![[rooted-tree.png]]
Back: $3$, $10$, and $4$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136845227 -->
END%%
%%ANKI
Basic
What are the internal nodes of the following rooted tree?
![[rooted-tree.png]]
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).
<!-- ID: 1711136845231 -->
END%%
%%ANKI
Basic
What are the external nodes of the following rooted tree?
![[rooted-tree.png]]
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).
<!-- ID: 1711136845235 -->
END%%
%%ANKI
Basic
What level does node $6$ reside on in the following rooted tree?
![[rooted-tree.png]]
Back: $3$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1711136845240 -->
END%%
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### Ordered Trees
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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).
<!-- ID: 1712406878904 -->
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).
<!-- ID: 1712406878909 -->
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).
<!-- ID: 1712406878912 -->
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).
<!-- ID: 1712406878915 -->
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).
<!-- ID: 1712406878917 -->
END%%
%%ANKI
Basic
The following two trees are equivalent when considered as what kind of trees?
![[ordered-rooted-tree-cmp.png]]
Back: Rooted/free trees.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1712407152755 -->
END%%
%%ANKI
Basic
The following two trees are different when considered as what kind of trees?
![[ordered-rooted-tree-cmp.png]]
Back: Ordered trees.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1712407152763 -->
END%%
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%%ANKI
Basic
Considered as rooted trees, are the following trees the same?
![[ordered-binary-tree-cmp.png]]
Back: Yes.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1712409466660 -->
END%%
%%ANKI
Basic
Considered as ordered trees, are the following trees the same?
![[ordered-binary-tree-cmp.png]]
Back: Yes.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1712409466670 -->
END%%
%%ANKI
Basic
Considered as binary trees, are the following trees the same?
![[ordered-binary-tree-cmp.png]]
Back: No.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1712409466676 -->
END%%
### Binary Trees
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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).
<!-- ID: 1712409466593 -->
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).
<!-- ID: 1712409466606 -->
END%%
%%ANKI
Basic
How should the nil constructor of an inductive binary tree, say `Tree` , be defined?
Back:
```lean
| constructor : Tree α
```
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: lean
<!-- ID: 1712409466615 -->
END%%
%%ANKI
Basic
How should the non-nil constructor of an inductive binary tree, say `Tree` , be defined?
Back:
```lean
| constructor : α α α α
```
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: lean
<!-- ID: 1712409466621 -->
END%%
%%ANKI
Basic
In the following binary tree type, what name is given to the first argument of `node` ?
```lean
inductive Tree α
| nil : Tree α
| node : α α α α
```
Back: The root node.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: lean
<!-- ID: 1712409466627 -->
END%%
%%ANKI
Basic
In the following binary tree type, what name is given to the second argument of `node` ?
```lean
inductive Tree α
| nil : Tree α
| node : α α α α
```
Back: The left subtree.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: lean
<!-- ID: 1712409466634 -->
END%%
%%ANKI
Basic
In the following binary tree type, what name is given to the third argument of `node` ?
```lean
inductive Tree α
| nil : Tree α
| node : α α α α
```
Back: The right subtree.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: lean
<!-- ID: 1712409466639 -->
END%%
%%ANKI
Basic
Given the following binary tree implementation, how do you construct an empty tree?
```lean
inductive Tree α
| nil : Tree α
| node : α α α α
```
Back: `nil`
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: lean
<!-- ID: 1712409466643 -->
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` ?
```lean
inductive Tree α
| nil : Tree α
| node : α α α 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
<!-- ID: 1712409466648 -->
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).
<!-- ID: 1712409466653 -->
END%%
%%ANKI
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Is a binary tree a $k$-ary tree?
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Back: Yes.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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END%%
ANKI%%
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%%
ANK%%
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).
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END%%
%%ANKI
Basic
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Why are these two binary trees not the same?
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![[ordered-binary-tree-cmp.png]]
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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).
<!-- ID: 1712409466682 -->
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 **complete** if all leaves have the same depth and all internal nodes have degree $k$.
%%ANKI
Basic
What does it mean for a binary tree to be full?
Back: Each node has $0$ or $2$ children.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1713118128213 -->
END%%
%%ANKI
What does it mean for a binary tree to be complete?
Back: Each leaf has the same depth and all internal nodes have degree $2$.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
What degrees are permitted in a full binary tree?
Back: $0$ or $2$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
What degrees are permitted in a complete binary tree?
Back: $0$ or $2$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
What category of rooted tree does a binary tree fall under?
Back: A positional tree or $k$-ary tree.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI
Basic
What distinguishes a positional tree from a $k$-ary tree?
Back: A $k$-ary tree is a positional tree in which each node has at most $k$ children.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1713118128216 -->
END%%
%%ANKI
Basic
What distinguishes positional trees from ordered trees?
Back: The same children in different positions is considered distinct in the former.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1713118128219 -->
END%%
%%ANKI
Basic
What is a positional tree?
Back: A rooted tree in which each child is labeled with a specific positive integer.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1713118128220 -->
END%%
%%ANKI
Basic
What is a $k$-ary tree?
Back: A positional tree in which each node has at most $k$ children.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1713118128223 -->
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).
<!-- ID: 1713118128225 -->
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
Is a binary tree a positional tree?
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Back: Yes.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!-- ID: 1713118128227 -->
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).
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END%%
%%ANKI
Basic
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Is the concept of completeness related to positional trees or $k$-ary trees?
Back: $k$-ary trees.
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Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!-- ID: 1713118128229 -->
2024-04-14 17:58:01 +00:00
END%%
%%ANKI
Basic
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What does it mean for a $k$-ary tree to be full?
Back: Each node has $0$ or $k$ children.
2024-04-14 17:58:01 +00:00
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
2024-04-14 18:44:20 +00:00
<!-- ID: 1713118128231 -->
END%%
%%ANKI
Basic
What degrees are permitted in a full $k$-ary tree?
Back: $0$ or $k$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1713118128233 -->
END%%
%%ANKI
Basic
What degrees are permitted in a complete $k$-ary tree?
Back: $0$ or $k$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1713118128234 -->
END%%
%%ANKI
Basic
What does it mean for a $k$-ary tree to be complete?
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).
<!-- ID: 1713118128236 -->
END%%
%%ANKI
Basic
What is the degree of an internal node in a complete $k$-ary tree'?
Back: $k$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1713118128239 -->
END%%
%%ANKI
Basic
What is the degree of an external node in a complete $k$-ary tree'?
Back: $0$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!-- ID: 1713118128241 -->
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).
<!-- ID: 1713118128242 -->
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).
<!-- ID: 1713118128244 -->
END%%
%%ANKI
Basic
What recursive definition describes the number of nodes in each level of a complete $k$-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
<!-- ID: 1713118128248 -->
END%%
%%ANKI
Basic
What closed formula details the number of nodes in a complete $k$-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
<!-- ID: 1713118128249 -->
END%%
%%ANKI
Basic
What kind of sequence describes the number of nodes in a complete $k$-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
<!-- ID: 1713118128251 -->
END%%
%%ANKI
Basic
What is the common ratio in the geometric sequence counting nodes of a complete $k$-ary tree?
Back: $k$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: algebra::sequence
<!-- ID: 1713118128253 -->
END%%
%%ANKI
Basic
How many nodes are in a complete binary tree of height $h$?
Back: $$\frac{1 - 2^h}{1 - 2} = 2^h - 1$$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: algebra::sequence
<!-- ID: 1713118128255 -->
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END%%
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## Bibliography
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* Thomas H. Cormen et al., _Introduction to Algorithms_ , Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
