notebook/notes/set/trees.md

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

## Overview

A **free tree** is a connected, acyclic, undirected [[set/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).
<!--ID: 1711136844897-->
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. It is.
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).
<!--ID: 1711136844943-->
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 [[set/graphs#Paths|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 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 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
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
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).
<!--ID: 1711136844962-->
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).
<!--ID: 1711136844969-->
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?
Back: The 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).
<!--ID: 1711136844980-->
END%%

%%ANKI
Basic
What does it mean for node $y$ to be a descendent of node $x$ in a rooted tree?
Back: The 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).
<!--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 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?
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).
<!--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 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 $h$?
Back: The external nodes on the longest paths from the root to said nodes.
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 of $4$ 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%%

### 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).
<!--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 (most specific) kind of trees?
![[ordered-rooted-tree-cmp.png]]
Back: Rooted 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 (most general) 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%%

%%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 positional 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: 1714089436122-->
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%%

%%ANKI
Basic
Why are these two binary trees not the same?
![[ordered-binary-tree-cmp.png]]
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%%

%%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).
<!--ID: 1715969047043-->
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).
<!--ID: 1715969047046-->
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).
<!--ID: 1715969047047-->
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).
<!--ID: 1715969047049-->
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).
<!--ID: 1715969047051-->
END%%

%%ANKI
Basic
How is a `struct` of a $k$-child tree representation written?
Back:
```c
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
<!--ID: 1715969047052-->
END%%

%%ANKI
Basic
What tree representation corresponds to the following `struct`?
```c
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
<!--ID: 1715969047054-->
END%%

%%ANKI
Basic
How is a `struct` of a left-child, right-sibling tree representation written?
Back:
```c
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
<!--ID: 1715969047056-->
END%%

%%ANKI
Basic
What tree representation corresponds to the following `struct`?
```c
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
<!--ID: 1715969047057-->
END%%

%%ANKI
Basic
What is an LCRS tree representation?
Back: A **l**eft-**c**hild, **r**ight-**s**ibling representation.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1715969525815-->
END%%

%%ANKI
Basic
The following is a portion of what kind of tree representation?
![[lcrs-nodes.png]]
Back: A left-child, right-sibling representation.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1715969525819-->
END%%

%%ANKI
Basic
The following is a portion of what kind of tree representation?
![[binary-tree-nodes.png]]
Back: A $k$-child (binary) representation.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1715969525820-->
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|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](https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees).
<!--ID: 1714088438740-->
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).
<!--ID: 1713118128216-->
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).
<!--ID: 1714089436130-->
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).
<!--ID: 1714089436134-->
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).
<!--ID: 1713118128219-->
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).
<!--ID: 1714088438749-->
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).
<!--ID: 1714088438754-->
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).
<!--ID: 1714088438759-->
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).
<!--ID: 1713118128220-->
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).
<!--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 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).
<!--ID: 1714088438763-->
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).
<!--ID: 1713118128229-->
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).
<!--ID: 1714088723844-->
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).
<!--ID: 1713118128231-->
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).
<!--ID: 1713118128233-->
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).
<!--ID: 1713118128234-->
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).
<!--ID: 1713118128236-->
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).
<!--ID: 1713118128239-->
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).
<!--ID: 1713118128241-->
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
<!--ID: 1713118128248-->
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
<!--ID: 1713118128249-->
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
<!--ID: 1714080353459-->
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
<!--ID: 1714080353455-->
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
<!--ID: 1714080353462-->
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
<!--ID: 1713118128251-->
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
<!--ID: 1713118128253-->
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).
<!--ID: 1714080353480-->
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).
<!--ID: 1714082676018-->
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).
<!--ID: 1714082676014-->
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).
<!--ID: 1714082676022-->
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).
<!--ID: 1714349367630-->
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).