417 lines
13 KiB
Markdown
417 lines
13 KiB
Markdown
---
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title: Binary Tree
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TARGET DECK: Obsidian::STEM
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FILE TAGS: data_structure::bst
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tags:
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- data_structure
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- graph
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- tree
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---
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## Overview
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A **binary tree** $T$ is a structure defined on a finite set of nodes that either
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* contains no nodes, or
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* is composed of three disjoint sets of nodes: a **root** node, a **left subtree**, and a **right subtree**.
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The binary tree is a specialization of the [[trees#Positional Trees|k-ary tree]].
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%%ANKI
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Basic
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Is a binary tree a $k$-ary 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: 1714089436138-->
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END%%
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%%ANKI
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Basic
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Is a binary tree a positional 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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END%%
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%%ANKI
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Basic
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Is a binary tree an ordered 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: 1714089436144-->
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END%%
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%%ANKI
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Basic
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What does it mean for a binary tree to be full?
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Back: Each node has $0$ or $2$ 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: 1713118128213-->
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END%%
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%%ANKI
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Basic
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What does it mean for a binary tree to be perfect?
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Back: Each leaf has the same depth and all internal nodes have degree $2$.
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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: 1714081594570-->
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END%%
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%%ANKI
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Basic
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Is a perfect binary tree considered full?
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Back: Yes.
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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).
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<!--ID: 1714088438720-->
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END%%
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%%ANKI
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Basic
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Is a full binary tree considered perfect?
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Back: Not necessarily.
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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).
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<!--ID: 1714088438726-->
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END%%
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%%ANKI
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Basic
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Is a full binary tree considered complete?
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Back: Not necessarily.
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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).
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<!--ID: 1714088438729-->
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END%%
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%%ANKI
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Basic
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Is a complete binary tree considered full?
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Back: Not necessarily.
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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).
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<!--ID: 1714088438733-->
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END%%
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%%ANKI
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Basic
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What alternative term is sometimes used in favor of a "perfect binary tree"?
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Back: A "complete binary tree".
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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).
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<!--ID: 1714088438737-->
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END%%
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%%ANKI
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Basic
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What alternative term is sometimes used in favor over a "complete binary tree"?
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Back: Some authors may say "nearly complete" if the last level isn't completely filled.
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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).
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<!--ID: 1714088438744-->
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END%%
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%%ANKI
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Basic
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What degrees are permitted in a full binary tree?
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Back: $0$ or $2$
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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: 1714081594576-->
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END%%
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%%ANKI
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Basic
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What degrees are permitted in a perfect binary tree?
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Back: $0$ or $2$
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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: 1714081594579-->
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END%%
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%%ANKI
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Basic
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What category of rooted tree does a binary tree fall under?
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Back: A positional tree or $k$-ary 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: 1714081594582-->
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END%%
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%%ANKI
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Basic
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Is a binary tree a positional 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: 1713118128227-->
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END%%
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%%ANKI
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Basic
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How many nodes are in a perfect binary tree of height $h$?
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Back: $2^{h+1} - 1$
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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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Tags: algebra::sequence
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<!--ID: 1713118128255-->
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END%%
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%%ANKI
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Basic
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How many internal nodes are in a perfect binary tree of height $h$?
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Back: $2^h - 1$
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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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Tags: algebra::sequence
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<!--ID: 1714080353472-->
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END%%
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%%ANKI
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Basic
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How many external nodes are in a perfect binary tree of height $h$?
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Back: $2^h$
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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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Tags: algebra::sequence
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<!--ID: 1714080353469-->
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END%%
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%%ANKI
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Basic
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How many nodes are on level $d$ of a perfect binary tree?
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Back: $2^d$
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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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Tags: algebra::sequence
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<!--ID: 1714080353465-->
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END%%
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%%ANKI
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Basic
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How does the number of internal nodes compare to the number of external nodes in a perfect binary tree?
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Back: There is one more external node than internal 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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Tags: algebra::sequence
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<!--ID: 1714080353476-->
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END%%
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%%ANKI
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Basic
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Is the following a perfect binary tree?
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![[perfect-tree.png]]
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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: 1714180419777-->
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END%%
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%%ANKI
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Basic
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Is the following a complete binary tree?
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![[perfect-tree.png]]
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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: 1714180419781-->
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END%%
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%%ANKI
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Basic
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Is the following a full binary tree?
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![[perfect-tree.png]]
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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: 1714180419784-->
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END%%
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%%ANKI
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Basic
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Is the following a perfect binary tree?
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![[complete-tree.png]]
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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: 1714180419787-->
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END%%
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%%ANKI
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Basic
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Is the following a complete binary tree?
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![[complete-tree.png]]
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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: 1714180419789-->
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END%%
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%%ANKI
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Basic
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Is the following a full binary tree?
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![[complete-tree.png]]
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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: 1714180419793-->
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END%%
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%%ANKI
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Basic
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Is the following a perfect binary tree?
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![[non-complete-tree.png]]
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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: 1714180419802-->
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END%%
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%%ANKI
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Basic
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Is the following a complete binary tree?
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![[non-complete-tree.png]]
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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: 1714180419809-->
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END%%
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%%ANKI
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Basic
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Is the following a full binary tree?
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![[non-complete-tree.png]]
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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: 1714180419813-->
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END%%
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%%ANKI
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Basic
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What is the minimum number of nodes in a complete binary tree of height $h$?
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Back: $2^h$
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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: 1714082676010-->
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END%%
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%%ANKI
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Basic
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What is the base case used in the recursive definition of a binary tree?
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Back: The empty set.
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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: 1712409466593-->
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END%%
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%%ANKI
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Basic
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What recurrence is used in the recursive definition of a binary tree?
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Back: A binary tree is composed of a root node, a left subtree, and a right subtree.
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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: 1712409466606-->
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END%%
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%%ANKI
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Basic
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How should the nil constructor of an inductive binary tree, say `Tree`, be defined?
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Back:
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```lean
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| nil : Tree α
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```
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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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Tags: lean
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<!--ID: 1712409466615-->
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END%%
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%%ANKI
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Basic
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How should the non-nil constructor of an inductive binary tree, say `Tree`, be defined?
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Back:
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```lean
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| node : α → Tree α → Tree α → Tree α
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```
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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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Tags: lean
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<!--ID: 1712409466621-->
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END%%
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%%ANKI
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Basic
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In the following binary tree type, what name is given to the first argument of `node`?
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```lean
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inductive Tree α where
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| nil : Tree α
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| node : α → Tree α → Tree α → Tree α
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```
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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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Tags: lean
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<!--ID: 1712409466627-->
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END%%
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%%ANKI
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Basic
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In the following binary tree type, what name is given to the second argument of `node`?
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```lean
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inductive Tree α where
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| nil : Tree α
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| node : α → Tree α → Tree α → Tree α
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```
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Back: The left subtree.
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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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Tags: lean
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<!--ID: 1712409466634-->
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END%%
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%%ANKI
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Basic
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In the following binary tree type, what name is given to the third argument of `node`?
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```lean
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inductive Tree α where
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| nil : Tree α
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| node : α → Tree α → Tree α → Tree α
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```
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Back: The right subtree.
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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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Tags: lean
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<!--ID: 1712409466639-->
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END%%
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%%ANKI
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Basic
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Given the following binary tree implementation, how do you construct an empty tree?
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```lean
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inductive Tree α where
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| nil : Tree α
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| node : α → Tree α → Tree α → Tree α
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```
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Back: `nil`
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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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Tags: lean
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<!--ID: 1712409466643-->
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END%%
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%%ANKI
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Basic
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Given the following binary tree implementation, how do you construct a tree with root `a`, left child `b`, and right child `c`?
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```lean
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inductive Tree α where
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| nil : Tree α
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| node : α → Tree α → Tree α → Tree α
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```
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Back: `node 'a' (node 'b' nil nil) (node 'c' nil nil)`
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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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Tags: lean
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<!--ID: 1712409466648-->
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END%%
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%%ANKI
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Basic
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Why isn't a binary tree considered an ordered tree?
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Back: A left child is distinct from a right child, even if the child is the same in both cases.
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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: 1712409466653-->
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END%%
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%%ANKI
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Basic
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How many internal nodes are in a complete binary tree of $n$ nodes?
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Back: $\lceil (n - 1) / 2 \rceil = \lfloor n / 2 \rfloor$
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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: 1714349367662-->
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END%%
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%%ANKI
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Basic
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A node of a binary tree typically has what three pointers?
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Back: The parent, left child, and right 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: 1715969047059-->
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END%%
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%%ANKI
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Basic
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In what direction do binary trees grow?
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Back: New nodes are added to the bottom of 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: 1723208264398-->
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END%%
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## Bibliography
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* “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).
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* Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). |