1061 lines
34 KiB
Markdown
1061 lines
34 KiB
Markdown
---
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title: Trees
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TARGET DECK: Obsidian::STEM
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FILE TAGS: set::graph
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tags:
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- graph
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- set
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---
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## Overview
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A **free tree** is a connected, acyclic, undirected [[graphs|graph]]. If an undirected graph is acyclic but possibly disconnected, it is a **forest**.
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%%ANKI
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Basic
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What is a free tree?
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Back: A connected, acyclic, undirected graph.
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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: 1711136844897-->
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END%%
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%%ANKI
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Basic
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What is a forest?
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Back: An acyclic undirected graph.
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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: 1711136844903-->
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END%%
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%%ANKI
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Basic
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What additional property must an undirected graph exhibit to be a forest?
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Back: It must be acyclic.
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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: 1711136844906-->
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END%%
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%%ANKI
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Basic
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What additional properties must an undirected graph exhibit to be a free tree?
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Back: It must be acyclic and connected.
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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: 1711136844909-->
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END%%
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%%ANKI
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Basic
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What additional properties must a forest exhibit to be a free tree?
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Back: It must be connected.
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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: 1711136844912-->
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END%%
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%%ANKI
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Basic
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What additional properties must a free tree exhibit to be a forest?
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Back: N/A
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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: 1711136844915-->
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END%%
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%%ANKI
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Basic
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If the following isn't a free tree, why not?
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![[free-tree.png]]
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Back: N/A
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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: 1711136844918-->
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END%%
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%%ANKI
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Basic
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If the following isn't a free tree, why not?
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![[forest.png]]
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Back: Because it is disconnected.
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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: 1711136844922-->
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END%%
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%%ANKI
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Basic
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If the following isn't a free tree, why not?
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![[cyclic-undirected.png]]
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Back: Because it contains a cycle.
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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: 1711136844926-->
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END%%
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%%ANKI
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Basic
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If the following isn't a forest, why not?
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![[free-tree.png]]
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Back: N/A
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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: 1711136844930-->
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END%%
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%%ANKI
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Basic
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If the following isn't a forest, why not?
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![[forest.png]]
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Back: N/A
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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: 1711136844934-->
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END%%
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%%ANKI
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Basic
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If the following isn't a forest, why not?
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![[cyclic-undirected.png]]
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Back: Because it contains a cycle.
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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: 1711136844939-->
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END%%
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%%ANKI
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Basic
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How do free trees pictorially relate to forests?
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Back: A forest is drawn as one or more free 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: 1711136844943-->
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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**.
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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**.
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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.
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%%ANKI
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Basic
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What is a rooted tree?
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Back: A free tree in which one of the vertices is distinguished from the others.
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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: 1711136844947-->
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END%%
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%%ANKI
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Basic
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Is every rooted tree a free tree?
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Back: Yes.
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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: 1711136844951-->
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END%%
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%%ANKI
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Basic
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Is every free tree a rooted tree?
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Back: No.
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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: 1711136844955-->
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END%%
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%%ANKI
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Basic
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Which free trees are not considered rooted trees?
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Back: Those without some vertex identified as the root.
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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: 1711136844958-->
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END%%
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%%ANKI
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Basic
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What distinguishes a node from a vertex?
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Back: A node is a vertex of a rooted tree.
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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: 1711136844962-->
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END%%
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%%ANKI
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Basic
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Is every vertex a node?
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Back: No.
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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: 1711136844966-->
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END%%
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%%ANKI
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Basic
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Is every node a vertex?
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Back: Yes.
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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: 1711136844969-->
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END%%
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%%ANKI
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Cloze
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{Nodes} are to rooted trees whereas {vertices} are to free 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: 1711136844973-->
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END%%
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%%ANKI
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Basic
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Which of free trees or rooted trees is a more general concept?
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Back: Free 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: 1711136844976-->
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END%%
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%%ANKI
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Basic
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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).
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<!--ID: 1711136844980-->
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END%%
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%%ANKI
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Basic
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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).
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<!--ID: 1711136844983-->
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END%%
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%%ANKI
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Cloze
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In a rooted tree, if $y$ is an {ancestor} of $x$, then $x$ is a {descendant} of $y$.
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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: 1711136844986-->
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END%%
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%%ANKI
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Basic
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What are the ancestors of a rooted tree's root?
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Back: Just the root itself.
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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: 1711136844989-->
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END%%
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%%ANKI
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Basic
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What are the descendants of a rooted tree's root?
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Back: Every node in the tree.
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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: 1711136844993-->
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END%%
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%%ANKI
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Basic
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What are the proper ancestors of a rooted tree's root?
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Back: There are none.
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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: 1711136844996-->
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END%%
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%%ANKI
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Basic
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What are the proper descendants of a rooted tree's root?
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Back: Every node but the root.
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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: 1711136845000-->
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END%%
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%%ANKI
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Basic
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What does it mean for node $y$ to be a child of node $x$ in a rooted tree?
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Back: There exists a path from the root to $y$ such that the last edge is $\{x, y\}$.
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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: 1711136845004-->
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END%%
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%%ANKI
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Basic
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What does it mean for node $y$ to be a parent of node $x$ in a rooted tree?
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Back: There exists a path from the root to $x$ such that the last edge is $\{y, x\}$.
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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: 1711136845009-->
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END%%
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%%ANKI
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Basic
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In a rooted tree, how does the concept of "ancestor" relate to "parent"?
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Back: Ancestors include parents, parents of parents, etc.
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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: 1711136845015-->
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END%%
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%%ANKI
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Basic
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In a rooted tree, how does the concept of "descendants" relate to "child"?
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Back: Descendants include children, children of children, etc.
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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: 1711136845020-->
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END%%
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%%ANKI
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Basic
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In a rooted tree, how many ancestors does a node have?
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Back: At least one (i.e. itself).
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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: 1711136845026-->
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END%%
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%%ANKI
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Basic
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In a rooted tree, how many parents does a node have?
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Back: Zero or one.
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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: 1711136845031-->
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END%%
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%%ANKI
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Basic
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In a rooted tree, how many descendants does a node have?
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Back: At least one (i.e. itself).
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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: 1711136845037-->
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END%%
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%%ANKI
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Basic
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In a rooted tree, how many children does a node have?
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Back: Zero or more.
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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: 1711136845044-->
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END%%
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%%ANKI
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Basic
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Which nodes in a rooted tree has no parent?
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Back: Just the root.
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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: 1711136845051-->
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END%%
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%%ANKI
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Basic
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In a rooted tree, what are siblings?
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Back: Nodes that have the same parent.
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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: 1711136845057-->
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END%%
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%%ANKI
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Basic
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In a rooted tree, what is an external node?
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Back: A node with no children.
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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: 1711136845063-->
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END%%
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%%ANKI
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Basic
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In a rooted tree, what alternative term is used in favor of "external node"?
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Back: A leaf.
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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: 1711136845072-->
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END%%
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%%ANKI
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Basic
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In a rooted tree, what is an internal node?
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Back: A node with at least one child.
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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: 1711136845079-->
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END%%
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%%ANKI
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Basic
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In a rooted tree, what alternative term is used in favor of "internal node"?
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Back: A nonleaf.
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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: 1711136845087-->
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END%%
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%%ANKI
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Cloze
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{1:External} nodes are to {2:leaf} nodes whereas {2:internal} nodes are to {1:nonleaf} nodes.
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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: 1711136845093-->
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END%%
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%%ANKI
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Basic
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Let $T$ be a rooted tree. What does the degree of a node refer to?
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Back: The number of children that node has.
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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: 1711136845101-->
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END%%
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%%ANKI
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Basic
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Let $T$ be a rooted tree. What does the depth of a node refer to?
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Back: The length of the simple path from the root to the node.
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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: 1711136845107-->
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END%%
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%%ANKI
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Basic
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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).
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<!--ID: 1711136845114-->
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END%%
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%%ANKI
|
||
Basic
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Let $T$ be a rooted tree. What does the height of a node refer to?
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Back: The length of the longest simple path from said node to a leaf.
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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: 1711136845119-->
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END%%
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%%ANKI
|
||
Basic
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What is the height of a rooted tree in terms of "height"?
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Back: The height of its root.
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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: 1711136845124-->
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END%%
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||
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%%ANKI
|
||
Basic
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What is the height of a rooted tree in terms of "depth"?
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Back: The largest depth of any node in the tree.
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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: 1711136845131-->
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END%%
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%%ANKI
|
||
Basic
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||
Let $T$ be a rooted tree of height $h$. Which nodes have height $0$?
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Back: The external nodes.
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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: 1711136845137-->
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END%%
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%%ANKI
|
||
Basic
|
||
Let $T$ be a rooted tree of height $h$. Which nodes have height $h$?
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Back: The root node.
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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: 1711136845141-->
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END%%
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%%ANKI
|
||
Basic
|
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Let $T$ be a rooted tree of height $h$. Which nodes have depth $0$?
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Back: The root.
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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: 1711136845145-->
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END%%
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%%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).
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<!--ID: 1711136845150-->
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END%%
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%%ANKI
|
||
Basic
|
||
What is the height of this rooted tree?
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||
![[rooted-tree.png]]
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Back: $4$
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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: 1711136845156-->
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END%%
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||
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%%ANKI
|
||
Basic
|
||
What is the height of node $4$ in the following rooted tree?
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||
![[rooted-tree.png]]
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Back: $1$
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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: 1711136845164-->
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||
END%%
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||
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||
%%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%%
|
||
|
||
### 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 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%%
|
||
|
||
%%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
|
||
|
||
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 → 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
|
||
<!--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 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
|
||
<!--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 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
|
||
<!--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 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
|
||
<!--ID: 1712409466639-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Given the following binary tree implementation, how do you construct an empty tree?
|
||
```lean
|
||
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
|
||
<!--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 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
|
||
<!--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
|
||
Is a binary tree a $k$-ary tree?
|
||
Back: Yes.
|
||
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
||
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).
|
||
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%%
|
||
|
||
### 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 $k$ labels with a potential child.
|
||
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?
|
||
Back: Yes.
|
||
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
||
<!--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).
|
||
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: 1713118128229-->
|
||
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: $0$ 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-->
|
||
END%%
|
||
|
||
## Bibliography
|
||
|
||
* Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|