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< span > Welcome to contribute to Chinese-to-English translation! Please visit < a href = "https://github.com/krahets/hello-algo/issues/914" > #914< / a > for more details.< / span >
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Chapter 0. Preface
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Chapter 0. Preface
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0.1 About this book
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0.2 How to read
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0.3 Summary
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Chapter 1. Encounter with algorithms
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Chapter 1. Encounter with algorithms
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1.1 Algorithms are everywhere
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1.2 What is an algorithm
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1.3 Summary
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Chapter 2. Complexity analysis
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Chapter 2. Complexity analysis
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2.1 Algorithm efficiency assessment
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2.2 Iteration and recursion
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2.3 Time complexity
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2.4 Space complexity
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2.5 Summary
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Chapter 3. Data structures
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Chapter 3. Data structures
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3.1 Classification of data structures
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3.2 Basic data types
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3.3 Number encoding *
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3.4 Character encoding *
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3.5 Summary
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Chapter 4. Array and linked list
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Chapter 4. Array and linked list
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4.1 Array
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4.2 Linked list
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4.3 List
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4.4 Memory and cache
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4.5 Summary
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Chapter 5. Stack and queue
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Chapter 5. Stack and queue
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5.1 Stack
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5.2 Queue
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5.3 Double-ended queue
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5.4 Summary
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Chapter 6. Hash table
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Chapter 6. Hash table
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6.1 Hash table
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6.2 Hash collision
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6.3 Hash algorithm
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6.4 Summary
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Chapter 7. Tree
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Chapter 7. Tree
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7.1 Binary tree
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7.1 Binary tree
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7.1.1 Common terminology of binary trees
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7.1.2 Basic operations of binary trees
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1. Initializing a binary tree
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2. Inserting and removing nodes
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7.1.3 Common types of binary trees
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1. Perfect binary tree
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2. Complete binary tree
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3. Full binary tree
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4. Balanced binary tree
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7.1.4 Degeneration of binary trees
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7.2 Binary tree Traversal
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7.3 Array Representation of tree
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7.4 Binary Search tree
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7.5 AVL tree *
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7.6 Summary
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Chapter 8. Heap
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8.1 Heap
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9.1 Graph
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Table of contents
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7.1.1 Common terminology of binary trees
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7.1.2 Basic operations of binary trees
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1. Initializing a binary tree
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7.1.3 Common types of binary trees
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4. Balanced binary tree
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7.1.4 Degeneration of binary trees
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< h1 id = "71-binary-tree" > 7.1 Binary tree< a class = "headerlink" href = "#71-binary-tree" title = "Permanent link" > ¶ < / a > < / h1 >
< p > A "binary tree" is a non-linear data structure that represents the ancestral and descendent relationships, embodying the "divide and conquer" logic. Similar to a linked list, the basic unit of a binary tree is a node, each containing a value, a reference to the left child node, and a reference to the right child node.< / p >
< div class = "tabbed-set tabbed-alternate" data-tabs = "1:14" > < input checked = "checked" id = "__tabbed_1_1" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_2" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_3" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_4" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_5" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_6" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_7" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_8" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_9" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_10" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_11" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_12" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_13" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_14" name = "__tabbed_1" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_1_1" > Python< / label > < label for = "__tabbed_1_2" > C++< / label > < label for = "__tabbed_1_3" > Java< / label > < label for = "__tabbed_1_4" > C#< / label > < label for = "__tabbed_1_5" > Go< / label > < label for = "__tabbed_1_6" > Swift< / label > < label for = "__tabbed_1_7" > JS< / label > < label for = "__tabbed_1_8" > TS< / label > < label for = "__tabbed_1_9" > Dart< / label > < label for = "__tabbed_1_10" > Rust< / label > < label for = "__tabbed_1_11" > C< / label > < label for = "__tabbed_1_12" > Kotlin< / label > < label for = "__tabbed_1_13" > Ruby< / label > < label for = "__tabbed_1_14" > Zig< / label > < / div >
< div class = "tabbed-content" >
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< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "k" > class< / span > < span class = "nc" > TreeNode< / span > < span class = "p" > :< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " Binary tree node" " " < / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "k" > def< / span > < span class = "fm" > __init__< / span > < span class = "p" > (< / span > < span class = "bp" > self< / span > < span class = "p" > ,< / span > < span class = "n" > val< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "bp" > self< / span > < span class = "o" > .< / span > < span class = "n" > val< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "o" > =< / span > < span class = "n" > val< / span > < span class = "c1" > # Node value< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "bp" > self< / span > < span class = "o" > .< / span > < span class = "n" > left< / span > < span class = "p" > :< / span > < span class = "n" > TreeNode< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "c1" > # Reference to left child node< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "bp" > self< / span > < span class = "o" > .< / span > < span class = "n" > right< / span > < span class = "p" > :< / span > < span class = "n" > TreeNode< / span > < span class = "o" > |< / span > < span class = "kc" > None< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "c1" > # Reference to right child node< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-1-1" name = "__codelineno-1-1" href = "#__codelineno-1-1" > < / a > < span class = "cm" > /* Binary tree node */< / span >
< a id = "__codelineno-1-2" name = "__codelineno-1-2" href = "#__codelineno-1-2" > < / a > < span class = "k" > struct< / span > < span class = "w" > < / span > < span class = "nc" > TreeNode< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-1-3" name = "__codelineno-1-3" href = "#__codelineno-1-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > val< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Node value< / span >
< a id = "__codelineno-1-4" name = "__codelineno-1-4" href = "#__codelineno-1-4" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "n" > left< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Pointer to left child node< / span >
< a id = "__codelineno-1-5" name = "__codelineno-1-5" href = "#__codelineno-1-5" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "n" > right< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Pointer to right child node< / span >
< a id = "__codelineno-1-6" name = "__codelineno-1-6" href = "#__codelineno-1-6" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > x< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "n" > val< / span > < span class = "p" > (< / span > < span class = "n" > x< / span > < span class = "p" > ),< / span > < span class = "w" > < / span > < span class = "n" > left< / span > < span class = "p" > (< / span > < span class = "k" > nullptr< / span > < span class = "p" > ),< / span > < span class = "w" > < / span > < span class = "n" > right< / span > < span class = "p" > (< / span > < span class = "k" > nullptr< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {}< / span >
< a id = "__codelineno-1-7" name = "__codelineno-1-7" href = "#__codelineno-1-7" > < / a > < span class = "p" > };< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-2-1" name = "__codelineno-2-1" href = "#__codelineno-2-1" > < / a > < span class = "cm" > /* Binary tree node */< / span >
< a id = "__codelineno-2-2" name = "__codelineno-2-2" href = "#__codelineno-2-2" > < / a > < span class = "kd" > class< / span > < span class = "nc" > TreeNode< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-2-3" name = "__codelineno-2-3" href = "#__codelineno-2-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > val< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Node value< / span >
< a id = "__codelineno-2-4" name = "__codelineno-2-4" href = "#__codelineno-2-4" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > left< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Reference to left child node< / span >
< a id = "__codelineno-2-5" name = "__codelineno-2-5" href = "#__codelineno-2-5" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > right< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Reference to right child node< / span >
< a id = "__codelineno-2-6" name = "__codelineno-2-6" href = "#__codelineno-2-6" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > x< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span > < span class = "w" > < / span > < span class = "n" > val< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > x< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-2-7" name = "__codelineno-2-7" href = "#__codelineno-2-7" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-3-1" name = "__codelineno-3-1" href = "#__codelineno-3-1" > < / a > < span class = "cm" > /* Binary tree node */< / span >
< a id = "__codelineno-3-2" name = "__codelineno-3-2" href = "#__codelineno-3-2" > < / a > < span class = "k" > class< / span > < span class = "w" > < / span > < span class = "nf" > TreeNode< / span > < span class = "p" > (< / span > < span class = "kt" > int?< / span > < span class = "w" > < / span > < span class = "n" > x< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-3-3" name = "__codelineno-3-3" href = "#__codelineno-3-3" > < / a > < span class = "w" > < / span > < span class = "k" > public< / span > < span class = "w" > < / span > < span class = "kt" > int?< / span > < span class = "w" > < / span > < span class = "n" > val< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > x< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Node value< / span >
< a id = "__codelineno-3-4" name = "__codelineno-3-4" href = "#__codelineno-3-4" > < / a > < span class = "w" > < / span > < span class = "k" > public< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "n" > left< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Reference to left child node< / span >
< a id = "__codelineno-3-5" name = "__codelineno-3-5" href = "#__codelineno-3-5" > < / a > < span class = "w" > < / span > < span class = "k" > public< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "n" > right< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Reference to right child node< / span >
< a id = "__codelineno-3-6" name = "__codelineno-3-6" href = "#__codelineno-3-6" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-4-1" name = "__codelineno-4-1" href = "#__codelineno-4-1" > < / a > < span class = "cm" > /* Binary tree node */< / span >
< a id = "__codelineno-4-2" name = "__codelineno-4-2" href = "#__codelineno-4-2" > < / a > < span class = "kd" > type< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "w" > < / span > < span class = "kd" > struct< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-4-3" name = "__codelineno-4-3" href = "#__codelineno-4-3" > < / a > < span class = "w" > < / span > < span class = "nx" > Val< / span > < span class = "w" > < / span > < span class = "kt" > int< / span >
< a id = "__codelineno-4-4" name = "__codelineno-4-4" href = "#__codelineno-4-4" > < / a > < span class = "w" > < / span > < span class = "nx" > Left< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "nx" > TreeNode< / span >
< a id = "__codelineno-4-5" name = "__codelineno-4-5" href = "#__codelineno-4-5" > < / a > < span class = "w" > < / span > < span class = "nx" > Right< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "nx" > TreeNode< / span >
< a id = "__codelineno-4-6" name = "__codelineno-4-6" href = "#__codelineno-4-6" > < / a > < span class = "p" > }< / span >
< a id = "__codelineno-4-7" name = "__codelineno-4-7" href = "#__codelineno-4-7" > < / a > < span class = "cm" > /* 构造方法 */< / span >
< a id = "__codelineno-4-8" name = "__codelineno-4-8" href = "#__codelineno-4-8" > < / a > < span class = "kd" > func< / span > < span class = "w" > < / span > < span class = "nx" > NewTreeNode< / span > < span class = "p" > (< / span > < span class = "nx" > v< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "nx" > TreeNode< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-4-9" name = "__codelineno-4-9" href = "#__codelineno-4-9" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "o" > & < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > {< / span >
< a id = "__codelineno-4-10" name = "__codelineno-4-10" href = "#__codelineno-4-10" > < / a > < span class = "w" > < / span > < span class = "nx" > Left< / span > < span class = "p" > :< / span > < span class = "w" > < / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "c1" > // Pointer to left child node< / span >
< a id = "__codelineno-4-11" name = "__codelineno-4-11" href = "#__codelineno-4-11" > < / a > < span class = "w" > < / span > < span class = "nx" > Right< / span > < span class = "p" > :< / span > < span class = "w" > < / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "c1" > // Pointer to right child node< / span >
< a id = "__codelineno-4-12" name = "__codelineno-4-12" href = "#__codelineno-4-12" > < / a > < span class = "w" > < / span > < span class = "nx" > Val< / span > < span class = "p" > :< / span > < span class = "w" > < / span > < span class = "nx" > v< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "c1" > // Node value< / span >
< a id = "__codelineno-4-13" name = "__codelineno-4-13" href = "#__codelineno-4-13" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-4-14" name = "__codelineno-4-14" href = "#__codelineno-4-14" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-5-1" name = "__codelineno-5-1" href = "#__codelineno-5-1" > < / a > < span class = "cm" > /* Binary tree node */< / span >
< a id = "__codelineno-5-2" name = "__codelineno-5-2" href = "#__codelineno-5-2" > < / a > < span class = "kd" > class< / span > < span class = "nc" > TreeNode< / span > < span class = "p" > {< / span >
< a id = "__codelineno-5-3" name = "__codelineno-5-3" href = "#__codelineno-5-3" > < / a > < span class = "kd" > var< / span > < span class = "nv" > val< / span > < span class = "p" > :< / span > < span class = "nb" > Int< / span > < span class = "c1" > // Node value< / span >
< a id = "__codelineno-5-4" name = "__codelineno-5-4" href = "#__codelineno-5-4" > < / a > < span class = "kd" > var< / span > < span class = "nv" > left< / span > < span class = "p" > :< / span > < span class = "n" > TreeNode< / span > < span class = "p" > ?< / span > < span class = "c1" > // Reference to left child node< / span >
< a id = "__codelineno-5-5" name = "__codelineno-5-5" href = "#__codelineno-5-5" > < / a > < span class = "kd" > var< / span > < span class = "nv" > right< / span > < span class = "p" > :< / span > < span class = "n" > TreeNode< / span > < span class = "p" > ?< / span > < span class = "c1" > // Reference to right child node< / span >
< a id = "__codelineno-5-6" name = "__codelineno-5-6" href = "#__codelineno-5-6" > < / a >
< a id = "__codelineno-5-7" name = "__codelineno-5-7" href = "#__codelineno-5-7" > < / a > < span class = "kd" > init< / span > < span class = "p" > (< / span > < span class = "n" > x< / span > < span class = "p" > :< / span > < span class = "nb" > Int< / span > < span class = "p" > )< / span > < span class = "p" > {< / span >
< a id = "__codelineno-5-8" name = "__codelineno-5-8" href = "#__codelineno-5-8" > < / a > < span class = "n" > val< / span > < span class = "p" > =< / span > < span class = "n" > x< / span >
< a id = "__codelineno-5-9" name = "__codelineno-5-9" href = "#__codelineno-5-9" > < / a > < span class = "p" > }< / span >
< a id = "__codelineno-5-10" name = "__codelineno-5-10" href = "#__codelineno-5-10" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-6-1" name = "__codelineno-6-1" href = "#__codelineno-6-1" > < / a > < span class = "cm" > /* Binary tree node */< / span >
< a id = "__codelineno-6-2" name = "__codelineno-6-2" href = "#__codelineno-6-2" > < / a > < span class = "kd" > class< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-6-3" name = "__codelineno-6-3" href = "#__codelineno-6-3" > < / a > < span class = "w" > < / span > < span class = "nx" > val< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Node value< / span >
< a id = "__codelineno-6-4" name = "__codelineno-6-4" href = "#__codelineno-6-4" > < / a > < span class = "w" > < / span > < span class = "nx" > left< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Pointer to left child node< / span >
< a id = "__codelineno-6-5" name = "__codelineno-6-5" href = "#__codelineno-6-5" > < / a > < span class = "w" > < / span > < span class = "nx" > right< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Pointer to right child node< / span >
< a id = "__codelineno-6-6" name = "__codelineno-6-6" href = "#__codelineno-6-6" > < / a > < span class = "w" > < / span > < span class = "kr" > constructor< / span > < span class = "p" > (< / span > < span class = "nx" > val< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "nx" > left< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "nx" > right< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-6-7" name = "__codelineno-6-7" href = "#__codelineno-6-7" > < / a > < span class = "w" > < / span > < span class = "k" > this< / span > < span class = "p" > .< / span > < span class = "nx" > val< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > val< / span > < span class = "w" > < / span > < span class = "o" > ===< / span > < span class = "w" > < / span > < span class = "kc" > undefined< / span > < span class = "w" > < / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "mf" > 0< / span > < span class = "w" > < / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "nx" > val< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-6-8" name = "__codelineno-6-8" href = "#__codelineno-6-8" > < / a > < span class = "w" > < / span > < span class = "k" > this< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > ===< / span > < span class = "w" > < / span > < span class = "kc" > undefined< / span > < span class = "w" > < / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "w" > < / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "nx" > left< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-6-9" name = "__codelineno-6-9" href = "#__codelineno-6-9" > < / a > < span class = "w" > < / span > < span class = "k" > this< / span > < span class = "p" > .< / span > < span class = "nx" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > right< / span > < span class = "w" > < / span > < span class = "o" > ===< / span > < span class = "w" > < / span > < span class = "kc" > undefined< / span > < span class = "w" > < / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "w" > < / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "nx" > right< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-6-10" name = "__codelineno-6-10" href = "#__codelineno-6-10" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-6-11" name = "__codelineno-6-11" href = "#__codelineno-6-11" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-7-1" name = "__codelineno-7-1" href = "#__codelineno-7-1" > < / a > < span class = "cm" > /* Binary tree node */< / span >
< a id = "__codelineno-7-2" name = "__codelineno-7-2" href = "#__codelineno-7-2" > < / a > < span class = "kd" > class< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-7-3" name = "__codelineno-7-3" href = "#__codelineno-7-3" > < / a > < span class = "w" > < / span > < span class = "nx" > val< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > number< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-7-4" name = "__codelineno-7-4" href = "#__codelineno-7-4" > < / a > < span class = "w" > < / span > < span class = "nx" > left< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > |< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-7-5" name = "__codelineno-7-5" href = "#__codelineno-7-5" > < / a > < span class = "w" > < / span > < span class = "nx" > right< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > |< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-7-6" name = "__codelineno-7-6" href = "#__codelineno-7-6" > < / a >
< a id = "__codelineno-7-7" name = "__codelineno-7-7" href = "#__codelineno-7-7" > < / a > < span class = "w" > < / span > < span class = "kr" > constructor< / span > < span class = "p" > (< / span > < span class = "nx" > val?< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > number< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "nx" > left?< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > |< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "nx" > right?< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > |< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-7-8" name = "__codelineno-7-8" href = "#__codelineno-7-8" > < / a > < span class = "w" > < / span > < span class = "k" > this< / span > < span class = "p" > .< / span > < span class = "nx" > val< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > val< / span > < span class = "w" > < / span > < span class = "o" > ===< / span > < span class = "w" > < / span > < span class = "kc" > undefined< / span > < span class = "w" > < / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "nx" > 0< / span > < span class = "w" > < / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > val< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Node value< / span >
< a id = "__codelineno-7-9" name = "__codelineno-7-9" href = "#__codelineno-7-9" > < / a > < span class = "w" > < / span > < span class = "k" > this< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > ===< / span > < span class = "w" > < / span > < span class = "kc" > undefined< / span > < span class = "w" > < / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "nx" > null< / span > < span class = "w" > < / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > left< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Reference to left child node< / span >
< a id = "__codelineno-7-10" name = "__codelineno-7-10" href = "#__codelineno-7-10" > < / a > < span class = "w" > < / span > < span class = "k" > this< / span > < span class = "p" > .< / span > < span class = "nx" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > right< / span > < span class = "w" > < / span > < span class = "o" > ===< / span > < span class = "w" > < / span > < span class = "kc" > undefined< / span > < span class = "w" > < / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "nx" > null< / span > < span class = "w" > < / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > right< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Reference to right child node< / span >
< a id = "__codelineno-7-11" name = "__codelineno-7-11" href = "#__codelineno-7-11" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-7-12" name = "__codelineno-7-12" href = "#__codelineno-7-12" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-8-1" name = "__codelineno-8-1" href = "#__codelineno-8-1" > < / a > < span class = "cm" > /* Binary tree node */< / span >
< a id = "__codelineno-8-2" name = "__codelineno-8-2" href = "#__codelineno-8-2" > < / a > < span class = "kd" > class< / span > < span class = "w" > < / span > < span class = "nc" > TreeNode< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-8-3" name = "__codelineno-8-3" href = "#__codelineno-8-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > val< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Node value< / span >
< a id = "__codelineno-8-4" name = "__codelineno-8-4" href = "#__codelineno-8-4" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "n" > left< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Reference to left child node< / span >
< a id = "__codelineno-8-5" name = "__codelineno-8-5" href = "#__codelineno-8-5" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "n" > right< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Reference to right child node< / span >
< a id = "__codelineno-8-6" name = "__codelineno-8-6" href = "#__codelineno-8-6" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "k" > this< / span > < span class = "p" > .< / span > < span class = "n" > val< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "p" > [< / span > < span class = "k" > this< / span > < span class = "p" > .< / span > < span class = "n" > left< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "k" > this< / span > < span class = "p" > .< / span > < span class = "n" > right< / span > < span class = "p" > ]);< / span >
< a id = "__codelineno-8-7" name = "__codelineno-8-7" href = "#__codelineno-8-7" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-9-1" name = "__codelineno-9-1" href = "#__codelineno-9-1" > < / a > < span class = "k" > use< / span > < span class = "w" > < / span > < span class = "n" > std< / span > ::< span class = "n" > rc< / span > ::< span class = "n" > Rc< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-9-2" name = "__codelineno-9-2" href = "#__codelineno-9-2" > < / a > < span class = "k" > use< / span > < span class = "w" > < / span > < span class = "n" > std< / span > ::< span class = "n" > cell< / span > ::< span class = "n" > RefCell< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-9-3" name = "__codelineno-9-3" href = "#__codelineno-9-3" > < / a >
< a id = "__codelineno-9-4" name = "__codelineno-9-4" href = "#__codelineno-9-4" > < / a > < span class = "cm" > /* Binary tree node */< / span >
< a id = "__codelineno-9-5" name = "__codelineno-9-5" href = "#__codelineno-9-5" > < / a > < span class = "k" > struct< / span > < span class = "nc" > TreeNode< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-9-6" name = "__codelineno-9-6" href = "#__codelineno-9-6" > < / a > < span class = "w" > < / span > < span class = "n" > val< / span > : < span class = "kt" > i32< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "c1" > // Node value< / span >
< a id = "__codelineno-9-7" name = "__codelineno-9-7" href = "#__codelineno-9-7" > < / a > < span class = "w" > < / span > < span class = "n" > left< / span > : < span class = "nb" > Option< / span > < span class = "o" > < < / span > < span class = "n" > Rc< / span > < span class = "o" > < < / span > < span class = "n" > RefCell< / span > < span class = "o" > < < / span > < span class = "n" > TreeNode< / span > < span class = "o" > > > > < / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "c1" > // Reference to left child node< / span >
< a id = "__codelineno-9-8" name = "__codelineno-9-8" href = "#__codelineno-9-8" > < / a > < span class = "w" > < / span > < span class = "n" > right< / span > : < span class = "nb" > Option< / span > < span class = "o" > < < / span > < span class = "n" > Rc< / span > < span class = "o" > < < / span > < span class = "n" > RefCell< / span > < span class = "o" > < < / span > < span class = "n" > TreeNode< / span > < span class = "o" > > > > < / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "c1" > // Reference to right child node< / span >
< a id = "__codelineno-9-9" name = "__codelineno-9-9" href = "#__codelineno-9-9" > < / a > < span class = "p" > }< / span >
< a id = "__codelineno-9-10" name = "__codelineno-9-10" href = "#__codelineno-9-10" > < / a >
< a id = "__codelineno-9-11" name = "__codelineno-9-11" href = "#__codelineno-9-11" > < / a > < span class = "k" > impl< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-9-12" name = "__codelineno-9-12" href = "#__codelineno-9-12" > < / a > < span class = "w" > < / span > < span class = "cm" > /* 构造方法 */< / span >
< a id = "__codelineno-9-13" name = "__codelineno-9-13" href = "#__codelineno-9-13" > < / a > < span class = "w" > < / span > < span class = "k" > fn< / span > < span class = "nf" > new< / span > < span class = "p" > (< / span > < span class = "n" > val< / span > : < span class = "kt" > i32< / span > < span class = "p" > )< / span > < span class = "w" > < / span > -> < span class = "nc" > Rc< / span > < span class = "o" > < < / span > < span class = "n" > RefCell< / span > < span class = "o" > < < / span > < span class = "bp" > Self< / span > < span class = "o" > > > < / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-9-14" name = "__codelineno-9-14" href = "#__codelineno-9-14" > < / a > < span class = "w" > < / span > < span class = "n" > Rc< / span > ::< span class = "n" > new< / span > < span class = "p" > (< / span > < span class = "n" > RefCell< / span > ::< span class = "n" > new< / span > < span class = "p" > (< / span > < span class = "bp" > Self< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-9-15" name = "__codelineno-9-15" href = "#__codelineno-9-15" > < / a > < span class = "w" > < / span > < span class = "n" > val< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-9-16" name = "__codelineno-9-16" href = "#__codelineno-9-16" > < / a > < span class = "w" > < / span > < span class = "n" > left< / span > : < span class = "nb" > None< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-9-17" name = "__codelineno-9-17" href = "#__codelineno-9-17" > < / a > < span class = "w" > < / span > < span class = "n" > right< / span > : < span class = "nb" > None< / span >
< a id = "__codelineno-9-18" name = "__codelineno-9-18" href = "#__codelineno-9-18" > < / a > < span class = "p" > }))< / span >
< a id = "__codelineno-9-19" name = "__codelineno-9-19" href = "#__codelineno-9-19" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-9-20" name = "__codelineno-9-20" href = "#__codelineno-9-20" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-10-1" name = "__codelineno-10-1" href = "#__codelineno-10-1" > < / a > < span class = "cm" > /* Binary tree node */< / span >
< a id = "__codelineno-10-2" name = "__codelineno-10-2" href = "#__codelineno-10-2" > < / a > < span class = "k" > typedef< / span > < span class = "w" > < / span > < span class = "k" > struct< / span > < span class = "w" > < / span > < span class = "nc" > TreeNode< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-10-3" name = "__codelineno-10-3" href = "#__codelineno-10-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > val< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Node value< / span >
< a id = "__codelineno-10-4" name = "__codelineno-10-4" href = "#__codelineno-10-4" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > height< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // 节点高度< / span >
< a id = "__codelineno-10-5" name = "__codelineno-10-5" href = "#__codelineno-10-5" > < / a > < span class = "w" > < / span > < span class = "k" > struct< / span > < span class = "w" > < / span > < span class = "nc" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "n" > left< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Pointer to left child node< / span >
< a id = "__codelineno-10-6" name = "__codelineno-10-6" href = "#__codelineno-10-6" > < / a > < span class = "w" > < / span > < span class = "k" > struct< / span > < span class = "w" > < / span > < span class = "nc" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "n" > right< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // Pointer to right child node< / span >
< a id = "__codelineno-10-7" name = "__codelineno-10-7" href = "#__codelineno-10-7" > < / a > < span class = "p" > }< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-10-8" name = "__codelineno-10-8" href = "#__codelineno-10-8" > < / a >
< a id = "__codelineno-10-9" name = "__codelineno-10-9" href = "#__codelineno-10-9" > < / a > < span class = "cm" > /* 构造函数 */< / span >
< a id = "__codelineno-10-10" name = "__codelineno-10-10" href = "#__codelineno-10-10" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "nf" > newTreeNode< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > val< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-10-11" name = "__codelineno-10-11" href = "#__codelineno-10-11" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "n" > node< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-10-12" name = "__codelineno-10-12" href = "#__codelineno-10-12" > < / a >
< a id = "__codelineno-10-13" name = "__codelineno-10-13" href = "#__codelineno-10-13" > < / a > < span class = "w" > < / span > < span class = "n" > node< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "p" > )< / span > < span class = "n" > malloc< / span > < span class = "p" > (< / span > < span class = "k" > sizeof< / span > < span class = "p" > (< / span > < span class = "n" > TreeNode< / span > < span class = "p" > ));< / span >
< a id = "__codelineno-10-14" name = "__codelineno-10-14" href = "#__codelineno-10-14" > < / a > < span class = "w" > < / span > < span class = "n" > node< / span > < span class = "o" > -> < / span > < span class = "n" > val< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > val< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-10-15" name = "__codelineno-10-15" href = "#__codelineno-10-15" > < / a > < span class = "w" > < / span > < span class = "n" > node< / span > < span class = "o" > -> < / span > < span class = "n" > height< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-10-16" name = "__codelineno-10-16" href = "#__codelineno-10-16" > < / a > < span class = "w" > < / span > < span class = "n" > node< / span > < span class = "o" > -> < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nb" > NULL< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-10-17" name = "__codelineno-10-17" href = "#__codelineno-10-17" > < / a > < span class = "w" > < / span > < span class = "n" > node< / span > < span class = "o" > -> < / span > < span class = "n" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nb" > NULL< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-10-18" name = "__codelineno-10-18" href = "#__codelineno-10-18" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > node< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-10-19" name = "__codelineno-10-19" href = "#__codelineno-10-19" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-11-1" name = "__codelineno-11-1" href = "#__codelineno-11-1" > < / a > < span class = "cm" > /* Binary tree node */< / span >
< a id = "__codelineno-11-2" name = "__codelineno-11-2" href = "#__codelineno-11-2" > < / a > < span class = "kd" > class< / span > < span class = "w" > < / span > < span class = "nc" > TreeNode< / span > < span class = "p" > (< / span > < span class = "kd" > val< / span > < span class = "w" > < / span > < span class = "nv" > _val< / span > < span class = "p" > :< / span > < span class = "w" > < / span > < span class = "kt" > Int< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span > < span class = "w" > < / span > < span class = "c1" > // Node value< / span >
< a id = "__codelineno-11-3" name = "__codelineno-11-3" href = "#__codelineno-11-3" > < / a > < span class = "w" > < / span > < span class = "kd" > val< / span > < span class = "w" > < / span > < span class = "nv" > left< / span > < span class = "p" > :< / span > < span class = "w" > < / span > < span class = "n" > TreeNode? < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "w" > < / span > < span class = "c1" > // Reference to left child node< / span >
< a id = "__codelineno-11-4" name = "__codelineno-11-4" href = "#__codelineno-11-4" > < / a > < span class = "w" > < / span > < span class = "kd" > val< / span > < span class = "w" > < / span > < span class = "nv" > right< / span > < span class = "p" > :< / span > < span class = "w" > < / span > < span class = "n" > TreeNode? < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "w" > < / span > < span class = "c1" > // Reference to right child node< / span >
< a id = "__codelineno-11-5" name = "__codelineno-11-5" href = "#__codelineno-11-5" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-12-1" name = "__codelineno-12-1" href = "#__codelineno-12-1" > < / a >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-13-1" name = "__codelineno-13-1" href = "#__codelineno-13-1" > < / a >
< / code > < / pre > < / div >
< / div >
< / div >
< / div >
< p > Each node has two references (pointers), pointing to the "left-child node" and "right-child node," respectively. This node is called the "parent node" of these two child nodes. When given a node of a binary tree, we call the tree formed by this node's left child and all nodes under it the "left subtree" of this node. Similarly, the "right subtree" can be defined.< / p >
< p > < strong > In a binary tree, except for leaf nodes, all other nodes contain child nodes and non-empty subtrees.< / strong > As shown in the Figure 7-1 , if "Node 2" is considered as the parent node, then its left and right child nodes are "Node 4" and "Node 5," respectively. The left subtree is "the tree formed by Node 4 and all nodes under it," and the right subtree is "the tree formed by Node 5 and all nodes under it."< / p >
< p > < a class = "glightbox" href = "../binary_tree.assets/binary_tree_definition.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Parent Node, child Node, subtree" class = "animation-figure" src = "../binary_tree.assets/binary_tree_definition.png" / > < / a > < / p >
< p align = "center" > Figure 7-1 Parent Node, child Node, subtree < / p >
< h2 id = "711-common-terminology-of-binary-trees" > 7.1.1 Common terminology of binary trees< a class = "headerlink" href = "#711-common-terminology-of-binary-trees" title = "Permanent link" > ¶ < / a > < / h2 >
< p > The commonly used terminology of binary trees is shown in the following figure.< / p >
< ul >
< li > "Root node": The node at the top level of the binary tree, which has no parent node.< / li >
< li > "Leaf node": A node with no children, both of its pointers point to < code > None< / code > .< / li >
< li > "Edge": The line segment connecting two nodes, i.e., node reference (pointer).< / li >
< li > The "level" of a node: Incrementing from top to bottom, with the root node's level being 1.< / li >
< li > The "degree" of a node: The number of a node's children. In a binary tree, the degree can be 0, 1, or 2.< / li >
< li > The "height" of a binary tree: The number of edges passed from the root node to the farthest leaf node.< / li >
< li > The "depth" of a node: The number of edges passed from the root node to the node.< / li >
< li > The "height" of a node: The number of edges from the farthest leaf node to the node.< / li >
< / ul >
< p > < a class = "glightbox" href = "../binary_tree.assets/binary_tree_terminology.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Common Terminology of Binary Trees" class = "animation-figure" src = "../binary_tree.assets/binary_tree_terminology.png" / > < / a > < / p >
< p align = "center" > Figure 7-2 Common Terminology of Binary Trees < / p >
< div class = "admonition tip" >
< p class = "admonition-title" > Tip< / p >
< p > Please note that we usually define "height" and "depth" as "the number of edges passed," but some problems or textbooks may define them as "the number of nodes passed." In this case, both height and depth need to be incremented by 1.< / p >
< / div >
< h2 id = "712-basic-operations-of-binary-trees" > 7.1.2 Basic operations of binary trees< a class = "headerlink" href = "#712-basic-operations-of-binary-trees" title = "Permanent link" > ¶ < / a > < / h2 >
< h3 id = "1-initializing-a-binary-tree" > 1. Initializing a binary tree< a class = "headerlink" href = "#1-initializing-a-binary-tree" title = "Permanent link" > ¶ < / a > < / h3 >
< p > Similar to a linked list, initialize nodes first, then construct references (pointers).< / p >
< div class = "tabbed-set tabbed-alternate" data-tabs = "2:14" > < input checked = "checked" id = "__tabbed_2_1" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_2" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_3" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_4" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_5" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_6" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_7" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_8" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_9" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_10" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_11" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_12" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_13" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_14" name = "__tabbed_2" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_2_1" > Python< / label > < label for = "__tabbed_2_2" > C++< / label > < label for = "__tabbed_2_3" > Java< / label > < label for = "__tabbed_2_4" > C#< / label > < label for = "__tabbed_2_5" > Go< / label > < label for = "__tabbed_2_6" > Swift< / label > < label for = "__tabbed_2_7" > JS< / label > < label for = "__tabbed_2_8" > TS< / label > < label for = "__tabbed_2_9" > Dart< / label > < label for = "__tabbed_2_10" > Rust< / label > < label for = "__tabbed_2_11" > C< / label > < label for = "__tabbed_2_12" > Kotlin< / label > < label for = "__tabbed_2_13" > Ruby< / label > < label for = "__tabbed_2_14" > Zig< / label > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-14-1" name = "__codelineno-14-1" href = "#__codelineno-14-1" > < / a > < span class = "c1" > # Initializing a binary tree< / span >
< a id = "__codelineno-14-2" name = "__codelineno-14-2" href = "#__codelineno-14-2" > < / a > < span class = "c1" > # Initializing nodes< / span >
< a id = "__codelineno-14-3" name = "__codelineno-14-3" href = "#__codelineno-14-3" > < / a > < span class = "n" > n1< / span > < span class = "o" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > val< / span > < span class = "o" > =< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span >
< a id = "__codelineno-14-4" name = "__codelineno-14-4" href = "#__codelineno-14-4" > < / a > < span class = "n" > n2< / span > < span class = "o" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > val< / span > < span class = "o" > =< / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span >
< a id = "__codelineno-14-5" name = "__codelineno-14-5" href = "#__codelineno-14-5" > < / a > < span class = "n" > n3< / span > < span class = "o" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > val< / span > < span class = "o" > =< / span > < span class = "mi" > 3< / span > < span class = "p" > )< / span >
< a id = "__codelineno-14-6" name = "__codelineno-14-6" href = "#__codelineno-14-6" > < / a > < span class = "n" > n4< / span > < span class = "o" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > val< / span > < span class = "o" > =< / span > < span class = "mi" > 4< / span > < span class = "p" > )< / span >
< a id = "__codelineno-14-7" name = "__codelineno-14-7" href = "#__codelineno-14-7" > < / a > < span class = "n" > n5< / span > < span class = "o" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > val< / span > < span class = "o" > =< / span > < span class = "mi" > 5< / span > < span class = "p" > )< / span >
< a id = "__codelineno-14-8" name = "__codelineno-14-8" href = "#__codelineno-14-8" > < / a > < span class = "c1" > # Linking references (pointers) between nodes< / span >
< a id = "__codelineno-14-9" name = "__codelineno-14-9" href = "#__codelineno-14-9" > < / a > < span class = "n" > n1< / span > < span class = "o" > .< / span > < span class = "n" > left< / span > < span class = "o" > =< / span > < span class = "n" > n2< / span >
< a id = "__codelineno-14-10" name = "__codelineno-14-10" href = "#__codelineno-14-10" > < / a > < span class = "n" > n1< / span > < span class = "o" > .< / span > < span class = "n" > right< / span > < span class = "o" > =< / span > < span class = "n" > n3< / span >
< a id = "__codelineno-14-11" name = "__codelineno-14-11" href = "#__codelineno-14-11" > < / a > < span class = "n" > n2< / span > < span class = "o" > .< / span > < span class = "n" > left< / span > < span class = "o" > =< / span > < span class = "n" > n4< / span >
< a id = "__codelineno-14-12" name = "__codelineno-14-12" href = "#__codelineno-14-12" > < / a > < span class = "n" > n2< / span > < span class = "o" > .< / span > < span class = "n" > right< / span > < span class = "o" > =< / span > < span class = "n" > n5< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.cpp< / span > < pre > < span > < / span > < code > < a id = "__codelineno-15-1" name = "__codelineno-15-1" href = "#__codelineno-15-1" > < / a > < span class = "cm" > /* Initializing a binary tree */< / span >
< a id = "__codelineno-15-2" name = "__codelineno-15-2" href = "#__codelineno-15-2" > < / a > < span class = "c1" > // Initializing nodes< / span >
< a id = "__codelineno-15-3" name = "__codelineno-15-3" href = "#__codelineno-15-3" > < / a > < span class = "n" > TreeNode< / span > < span class = "o" > *< / span > < span class = "w" > < / span > < span class = "n" > n1< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-15-4" name = "__codelineno-15-4" href = "#__codelineno-15-4" > < / a > < span class = "n" > TreeNode< / span > < span class = "o" > *< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 2< / span > < span class = "p" > );< / span >
< a id = "__codelineno-15-5" name = "__codelineno-15-5" href = "#__codelineno-15-5" > < / a > < span class = "n" > TreeNode< / span > < span class = "o" > *< / span > < span class = "w" > < / span > < span class = "n" > n3< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > );< / span >
< a id = "__codelineno-15-6" name = "__codelineno-15-6" href = "#__codelineno-15-6" > < / a > < span class = "n" > TreeNode< / span > < span class = "o" > *< / span > < span class = "w" > < / span > < span class = "n" > n4< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 4< / span > < span class = "p" > );< / span >
< a id = "__codelineno-15-7" name = "__codelineno-15-7" href = "#__codelineno-15-7" > < / a > < span class = "n" > TreeNode< / span > < span class = "o" > *< / span > < span class = "w" > < / span > < span class = "n" > n5< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 5< / span > < span class = "p" > );< / span >
< a id = "__codelineno-15-8" name = "__codelineno-15-8" href = "#__codelineno-15-8" > < / a > < span class = "c1" > // Linking references (pointers) between nodes< / span >
< a id = "__codelineno-15-9" name = "__codelineno-15-9" href = "#__codelineno-15-9" > < / a > < span class = "n" > n1< / span > < span class = "o" > -> < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-15-10" name = "__codelineno-15-10" href = "#__codelineno-15-10" > < / a > < span class = "n" > n1< / span > < span class = "o" > -> < / span > < span class = "n" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n3< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-15-11" name = "__codelineno-15-11" href = "#__codelineno-15-11" > < / a > < span class = "n" > n2< / span > < span class = "o" > -> < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n4< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-15-12" name = "__codelineno-15-12" href = "#__codelineno-15-12" > < / a > < span class = "n" > n2< / span > < span class = "o" > -> < / span > < span class = "n" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n5< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-16-1" name = "__codelineno-16-1" href = "#__codelineno-16-1" > < / a > < span class = "c1" > // Initializing nodes< / span >
< a id = "__codelineno-16-2" name = "__codelineno-16-2" href = "#__codelineno-16-2" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n1< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-16-3" name = "__codelineno-16-3" href = "#__codelineno-16-3" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 2< / span > < span class = "p" > );< / span >
< a id = "__codelineno-16-4" name = "__codelineno-16-4" href = "#__codelineno-16-4" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n3< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > );< / span >
< a id = "__codelineno-16-5" name = "__codelineno-16-5" href = "#__codelineno-16-5" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n4< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 4< / span > < span class = "p" > );< / span >
< a id = "__codelineno-16-6" name = "__codelineno-16-6" href = "#__codelineno-16-6" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n5< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 5< / span > < span class = "p" > );< / span >
< a id = "__codelineno-16-7" name = "__codelineno-16-7" href = "#__codelineno-16-7" > < / a > < span class = "c1" > // Linking references (pointers) between nodes< / span >
< a id = "__codelineno-16-8" name = "__codelineno-16-8" href = "#__codelineno-16-8" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "na" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-16-9" name = "__codelineno-16-9" href = "#__codelineno-16-9" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "na" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n3< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-16-10" name = "__codelineno-16-10" href = "#__codelineno-16-10" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "na" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n4< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-16-11" name = "__codelineno-16-11" href = "#__codelineno-16-11" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "na" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n5< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-17-1" name = "__codelineno-17-1" href = "#__codelineno-17-1" > < / a > < span class = "cm" > /* Initializing a binary tree */< / span >
< a id = "__codelineno-17-2" name = "__codelineno-17-2" href = "#__codelineno-17-2" > < / a > < span class = "c1" > // Initializing nodes< / span >
< a id = "__codelineno-17-3" name = "__codelineno-17-3" href = "#__codelineno-17-3" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n1< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "p" > (< / span > < span class = "m" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-17-4" name = "__codelineno-17-4" href = "#__codelineno-17-4" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "p" > (< / span > < span class = "m" > 2< / span > < span class = "p" > );< / span >
< a id = "__codelineno-17-5" name = "__codelineno-17-5" href = "#__codelineno-17-5" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n3< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "p" > (< / span > < span class = "m" > 3< / span > < span class = "p" > );< / span >
< a id = "__codelineno-17-6" name = "__codelineno-17-6" href = "#__codelineno-17-6" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n4< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "p" > (< / span > < span class = "m" > 4< / span > < span class = "p" > );< / span >
< a id = "__codelineno-17-7" name = "__codelineno-17-7" href = "#__codelineno-17-7" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n5< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "p" > (< / span > < span class = "m" > 5< / span > < span class = "p" > );< / span >
< a id = "__codelineno-17-8" name = "__codelineno-17-8" href = "#__codelineno-17-8" > < / a > < span class = "c1" > // Linking references (pointers) between nodes< / span >
< a id = "__codelineno-17-9" name = "__codelineno-17-9" href = "#__codelineno-17-9" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-17-10" name = "__codelineno-17-10" href = "#__codelineno-17-10" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "n" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n3< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-17-11" name = "__codelineno-17-11" href = "#__codelineno-17-11" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n4< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-17-12" name = "__codelineno-17-12" href = "#__codelineno-17-12" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "n" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n5< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-18-1" name = "__codelineno-18-1" href = "#__codelineno-18-1" > < / a > < span class = "cm" > /* Initializing a binary tree */< / span >
< a id = "__codelineno-18-2" name = "__codelineno-18-2" href = "#__codelineno-18-2" > < / a > < span class = "c1" > // Initializing nodes< / span >
< a id = "__codelineno-18-3" name = "__codelineno-18-3" href = "#__codelineno-18-3" > < / a > < span class = "nx" > n1< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nx" > NewTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span >
< a id = "__codelineno-18-4" name = "__codelineno-18-4" href = "#__codelineno-18-4" > < / a > < span class = "nx" > n2< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nx" > NewTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span >
< a id = "__codelineno-18-5" name = "__codelineno-18-5" href = "#__codelineno-18-5" > < / a > < span class = "nx" > n3< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nx" > NewTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > )< / span >
< a id = "__codelineno-18-6" name = "__codelineno-18-6" href = "#__codelineno-18-6" > < / a > < span class = "nx" > n4< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nx" > NewTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 4< / span > < span class = "p" > )< / span >
< a id = "__codelineno-18-7" name = "__codelineno-18-7" href = "#__codelineno-18-7" > < / a > < span class = "nx" > n5< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nx" > NewTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 5< / span > < span class = "p" > )< / span >
< a id = "__codelineno-18-8" name = "__codelineno-18-8" href = "#__codelineno-18-8" > < / a > < span class = "c1" > // Linking references (pointers) between nodes< / span >
< a id = "__codelineno-18-9" name = "__codelineno-18-9" href = "#__codelineno-18-9" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > Left< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span >
< a id = "__codelineno-18-10" name = "__codelineno-18-10" href = "#__codelineno-18-10" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > Right< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > n3< / span >
< a id = "__codelineno-18-11" name = "__codelineno-18-11" href = "#__codelineno-18-11" > < / a > < span class = "nx" > n2< / span > < span class = "p" > .< / span > < span class = "nx" > Left< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > n4< / span >
< a id = "__codelineno-18-12" name = "__codelineno-18-12" href = "#__codelineno-18-12" > < / a > < span class = "nx" > n2< / span > < span class = "p" > .< / span > < span class = "nx" > Right< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > n5< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.swift< / span > < pre > < span > < / span > < code > < a id = "__codelineno-19-1" name = "__codelineno-19-1" href = "#__codelineno-19-1" > < / a > < span class = "c1" > // Initializing nodes< / span >
< a id = "__codelineno-19-2" name = "__codelineno-19-2" href = "#__codelineno-19-2" > < / a > < span class = "kd" > let< / span > < span class = "nv" > n1< / span > < span class = "p" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > x< / span > < span class = "p" > :< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span >
< a id = "__codelineno-19-3" name = "__codelineno-19-3" href = "#__codelineno-19-3" > < / a > < span class = "kd" > let< / span > < span class = "nv" > n2< / span > < span class = "p" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > x< / span > < span class = "p" > :< / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span >
< a id = "__codelineno-19-4" name = "__codelineno-19-4" href = "#__codelineno-19-4" > < / a > < span class = "kd" > let< / span > < span class = "nv" > n3< / span > < span class = "p" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > x< / span > < span class = "p" > :< / span > < span class = "mi" > 3< / span > < span class = "p" > )< / span >
< a id = "__codelineno-19-5" name = "__codelineno-19-5" href = "#__codelineno-19-5" > < / a > < span class = "kd" > let< / span > < span class = "nv" > n4< / span > < span class = "p" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > x< / span > < span class = "p" > :< / span > < span class = "mi" > 4< / span > < span class = "p" > )< / span >
< a id = "__codelineno-19-6" name = "__codelineno-19-6" href = "#__codelineno-19-6" > < / a > < span class = "kd" > let< / span > < span class = "nv" > n5< / span > < span class = "p" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > x< / span > < span class = "p" > :< / span > < span class = "mi" > 5< / span > < span class = "p" > )< / span >
< a id = "__codelineno-19-7" name = "__codelineno-19-7" href = "#__codelineno-19-7" > < / a > < span class = "c1" > // Linking references (pointers) between nodes< / span >
< a id = "__codelineno-19-8" name = "__codelineno-19-8" href = "#__codelineno-19-8" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "kr" > left< / span > < span class = "p" > =< / span > < span class = "n" > n2< / span >
< a id = "__codelineno-19-9" name = "__codelineno-19-9" href = "#__codelineno-19-9" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "kr" > right< / span > < span class = "p" > =< / span > < span class = "n" > n3< / span >
< a id = "__codelineno-19-10" name = "__codelineno-19-10" href = "#__codelineno-19-10" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "kr" > left< / span > < span class = "p" > =< / span > < span class = "n" > n4< / span >
< a id = "__codelineno-19-11" name = "__codelineno-19-11" href = "#__codelineno-19-11" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "kr" > right< / span > < span class = "p" > =< / span > < span class = "n" > n5< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.js< / span > < pre > < span > < / span > < code > < a id = "__codelineno-20-1" name = "__codelineno-20-1" href = "#__codelineno-20-1" > < / a > < span class = "cm" > /* Initializing a binary tree */< / span >
< a id = "__codelineno-20-2" name = "__codelineno-20-2" href = "#__codelineno-20-2" > < / a > < span class = "c1" > // Initializing nodes< / span >
< a id = "__codelineno-20-3" name = "__codelineno-20-3" href = "#__codelineno-20-3" > < / a > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "nx" > n1< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 1< / span > < span class = "p" > ),< / span >
< a id = "__codelineno-20-4" name = "__codelineno-20-4" href = "#__codelineno-20-4" > < / a > < span class = "w" > < / span > < span class = "nx" > n2< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 2< / span > < span class = "p" > ),< / span >
< a id = "__codelineno-20-5" name = "__codelineno-20-5" href = "#__codelineno-20-5" > < / a > < span class = "w" > < / span > < span class = "nx" > n3< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 3< / span > < span class = "p" > ),< / span >
< a id = "__codelineno-20-6" name = "__codelineno-20-6" href = "#__codelineno-20-6" > < / a > < span class = "w" > < / span > < span class = "nx" > n4< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 4< / span > < span class = "p" > ),< / span >
< a id = "__codelineno-20-7" name = "__codelineno-20-7" href = "#__codelineno-20-7" > < / a > < span class = "w" > < / span > < span class = "nx" > n5< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 5< / span > < span class = "p" > );< / span >
< a id = "__codelineno-20-8" name = "__codelineno-20-8" href = "#__codelineno-20-8" > < / a > < span class = "c1" > // Linking references (pointers) between nodes< / span >
< a id = "__codelineno-20-9" name = "__codelineno-20-9" href = "#__codelineno-20-9" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-20-10" name = "__codelineno-20-10" href = "#__codelineno-20-10" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n3< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-20-11" name = "__codelineno-20-11" href = "#__codelineno-20-11" > < / a > < span class = "nx" > n2< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n4< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-20-12" name = "__codelineno-20-12" href = "#__codelineno-20-12" > < / a > < span class = "nx" > n2< / span > < span class = "p" > .< / span > < span class = "nx" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n5< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.ts< / span > < pre > < span > < / span > < code > < a id = "__codelineno-21-1" name = "__codelineno-21-1" href = "#__codelineno-21-1" > < / a > < span class = "cm" > /* Initializing a binary tree */< / span >
< a id = "__codelineno-21-2" name = "__codelineno-21-2" href = "#__codelineno-21-2" > < / a > < span class = "c1" > // Initializing nodes< / span >
< a id = "__codelineno-21-3" name = "__codelineno-21-3" href = "#__codelineno-21-3" > < / a > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "nx" > n1< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 1< / span > < span class = "p" > ),< / span >
< a id = "__codelineno-21-4" name = "__codelineno-21-4" href = "#__codelineno-21-4" > < / a > < span class = "w" > < / span > < span class = "nx" > n2< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 2< / span > < span class = "p" > ),< / span >
< a id = "__codelineno-21-5" name = "__codelineno-21-5" href = "#__codelineno-21-5" > < / a > < span class = "w" > < / span > < span class = "nx" > n3< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 3< / span > < span class = "p" > ),< / span >
< a id = "__codelineno-21-6" name = "__codelineno-21-6" href = "#__codelineno-21-6" > < / a > < span class = "w" > < / span > < span class = "nx" > n4< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 4< / span > < span class = "p" > ),< / span >
< a id = "__codelineno-21-7" name = "__codelineno-21-7" href = "#__codelineno-21-7" > < / a > < span class = "w" > < / span > < span class = "nx" > n5< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 5< / span > < span class = "p" > );< / span >
< a id = "__codelineno-21-8" name = "__codelineno-21-8" href = "#__codelineno-21-8" > < / a > < span class = "c1" > // Linking references (pointers) between nodes< / span >
< a id = "__codelineno-21-9" name = "__codelineno-21-9" href = "#__codelineno-21-9" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-21-10" name = "__codelineno-21-10" href = "#__codelineno-21-10" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n3< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-21-11" name = "__codelineno-21-11" href = "#__codelineno-21-11" > < / a > < span class = "nx" > n2< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n4< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-21-12" name = "__codelineno-21-12" href = "#__codelineno-21-12" > < / a > < span class = "nx" > n2< / span > < span class = "p" > .< / span > < span class = "nx" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n5< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.dart< / span > < pre > < span > < / span > < code > < a id = "__codelineno-22-1" name = "__codelineno-22-1" href = "#__codelineno-22-1" > < / a > < span class = "cm" > /* Initializing a binary tree */< / span >
< a id = "__codelineno-22-2" name = "__codelineno-22-2" href = "#__codelineno-22-2" > < / a > < span class = "c1" > // Initializing nodes< / span >
< a id = "__codelineno-22-3" name = "__codelineno-22-3" href = "#__codelineno-22-3" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n1< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-22-4" name = "__codelineno-22-4" href = "#__codelineno-22-4" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 2< / span > < span class = "p" > );< / span >
< a id = "__codelineno-22-5" name = "__codelineno-22-5" href = "#__codelineno-22-5" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n3< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 3< / span > < span class = "p" > );< / span >
< a id = "__codelineno-22-6" name = "__codelineno-22-6" href = "#__codelineno-22-6" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n4< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 4< / span > < span class = "p" > );< / span >
< a id = "__codelineno-22-7" name = "__codelineno-22-7" href = "#__codelineno-22-7" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n5< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 5< / span > < span class = "p" > );< / span >
< a id = "__codelineno-22-8" name = "__codelineno-22-8" href = "#__codelineno-22-8" > < / a > < span class = "c1" > // Linking references (pointers) between nodes< / span >
< a id = "__codelineno-22-9" name = "__codelineno-22-9" href = "#__codelineno-22-9" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-22-10" name = "__codelineno-22-10" href = "#__codelineno-22-10" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "n" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n3< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-22-11" name = "__codelineno-22-11" href = "#__codelineno-22-11" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n4< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-22-12" name = "__codelineno-22-12" href = "#__codelineno-22-12" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "n" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n5< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.rs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-23-1" name = "__codelineno-23-1" href = "#__codelineno-23-1" > < / a > < span class = "c1" > // Initializing nodes< / span >
< a id = "__codelineno-23-2" name = "__codelineno-23-2" href = "#__codelineno-23-2" > < / a > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "n" > n1< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > ::< span class = "n" > new< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-23-3" name = "__codelineno-23-3" href = "#__codelineno-23-3" > < / a > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > ::< span class = "n" > new< / span > < span class = "p" > (< / span > < span class = "mi" > 2< / span > < span class = "p" > );< / span >
< a id = "__codelineno-23-4" name = "__codelineno-23-4" href = "#__codelineno-23-4" > < / a > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "n" > n3< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > ::< span class = "n" > new< / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > );< / span >
< a id = "__codelineno-23-5" name = "__codelineno-23-5" href = "#__codelineno-23-5" > < / a > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "n" > n4< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > ::< span class = "n" > new< / span > < span class = "p" > (< / span > < span class = "mi" > 4< / span > < span class = "p" > );< / span >
< a id = "__codelineno-23-6" name = "__codelineno-23-6" href = "#__codelineno-23-6" > < / a > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "n" > n5< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > ::< span class = "n" > new< / span > < span class = "p" > (< / span > < span class = "mi" > 5< / span > < span class = "p" > );< / span >
< a id = "__codelineno-23-7" name = "__codelineno-23-7" href = "#__codelineno-23-7" > < / a > < span class = "c1" > // Linking references (pointers) between nodes< / span >
< a id = "__codelineno-23-8" name = "__codelineno-23-8" href = "#__codelineno-23-8" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "n" > borrow_mut< / span > < span class = "p" > ().< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nb" > Some< / span > < span class = "p" > (< / span > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "n" > clone< / span > < span class = "p" > ());< / span >
< a id = "__codelineno-23-9" name = "__codelineno-23-9" href = "#__codelineno-23-9" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "n" > borrow_mut< / span > < span class = "p" > ().< / span > < span class = "n" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nb" > Some< / span > < span class = "p" > (< / span > < span class = "n" > n3< / span > < span class = "p" > );< / span >
< a id = "__codelineno-23-10" name = "__codelineno-23-10" href = "#__codelineno-23-10" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "n" > borrow_mut< / span > < span class = "p" > ().< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nb" > Some< / span > < span class = "p" > (< / span > < span class = "n" > n4< / span > < span class = "p" > );< / span >
< a id = "__codelineno-23-11" name = "__codelineno-23-11" href = "#__codelineno-23-11" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "n" > borrow_mut< / span > < span class = "p" > ().< / span > < span class = "n" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nb" > Some< / span > < span class = "p" > (< / span > < span class = "n" > n5< / span > < span class = "p" > );< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-24-1" name = "__codelineno-24-1" href = "#__codelineno-24-1" > < / a > < span class = "cm" > /* Initializing a binary tree */< / span >
< a id = "__codelineno-24-2" name = "__codelineno-24-2" href = "#__codelineno-24-2" > < / a > < span class = "c1" > // Initializing nodes< / span >
< a id = "__codelineno-24-3" name = "__codelineno-24-3" href = "#__codelineno-24-3" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "n" > n1< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > newTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-24-4" name = "__codelineno-24-4" href = "#__codelineno-24-4" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "n" > n2< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > newTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 2< / span > < span class = "p" > );< / span >
< a id = "__codelineno-24-5" name = "__codelineno-24-5" href = "#__codelineno-24-5" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "n" > n3< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > newTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > );< / span >
< a id = "__codelineno-24-6" name = "__codelineno-24-6" href = "#__codelineno-24-6" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "n" > n4< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > newTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 4< / span > < span class = "p" > );< / span >
< a id = "__codelineno-24-7" name = "__codelineno-24-7" href = "#__codelineno-24-7" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "n" > n5< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > newTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 5< / span > < span class = "p" > );< / span >
< a id = "__codelineno-24-8" name = "__codelineno-24-8" href = "#__codelineno-24-8" > < / a > < span class = "c1" > // Linking references (pointers) between nodes< / span >
< a id = "__codelineno-24-9" name = "__codelineno-24-9" href = "#__codelineno-24-9" > < / a > < span class = "n" > n1< / span > < span class = "o" > -> < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-24-10" name = "__codelineno-24-10" href = "#__codelineno-24-10" > < / a > < span class = "n" > n1< / span > < span class = "o" > -> < / span > < span class = "n" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n3< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-24-11" name = "__codelineno-24-11" href = "#__codelineno-24-11" > < / a > < span class = "n" > n2< / span > < span class = "o" > -> < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n4< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-24-12" name = "__codelineno-24-12" href = "#__codelineno-24-12" > < / a > < span class = "n" > n2< / span > < span class = "o" > -> < / span > < span class = "n" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n5< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.kt< / span > < pre > < span > < / span > < code > < a id = "__codelineno-25-1" name = "__codelineno-25-1" href = "#__codelineno-25-1" > < / a > < span class = "c1" > // Initializing nodes< / span >
< a id = "__codelineno-25-2" name = "__codelineno-25-2" href = "#__codelineno-25-2" > < / a > < span class = "kd" > val< / span > < span class = "w" > < / span > < span class = "nv" > n1< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 1< / span > < span class = "p" > )< / span >
< a id = "__codelineno-25-3" name = "__codelineno-25-3" href = "#__codelineno-25-3" > < / a > < span class = "kd" > val< / span > < span class = "w" > < / span > < span class = "nv" > n2< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 2< / span > < span class = "p" > )< / span >
< a id = "__codelineno-25-4" name = "__codelineno-25-4" href = "#__codelineno-25-4" > < / a > < span class = "kd" > val< / span > < span class = "w" > < / span > < span class = "nv" > n3< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 3< / span > < span class = "p" > )< / span >
< a id = "__codelineno-25-5" name = "__codelineno-25-5" href = "#__codelineno-25-5" > < / a > < span class = "kd" > val< / span > < span class = "w" > < / span > < span class = "nv" > n4< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 4< / span > < span class = "p" > )< / span >
< a id = "__codelineno-25-6" name = "__codelineno-25-6" href = "#__codelineno-25-6" > < / a > < span class = "kd" > val< / span > < span class = "w" > < / span > < span class = "nv" > n5< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 5< / span > < span class = "p" > )< / span >
< a id = "__codelineno-25-7" name = "__codelineno-25-7" href = "#__codelineno-25-7" > < / a > < span class = "c1" > // Linking references (pointers) between nodes< / span >
< a id = "__codelineno-25-8" name = "__codelineno-25-8" href = "#__codelineno-25-8" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "na" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span >
< a id = "__codelineno-25-9" name = "__codelineno-25-9" href = "#__codelineno-25-9" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "na" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n3< / span >
< a id = "__codelineno-25-10" name = "__codelineno-25-10" href = "#__codelineno-25-10" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "na" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n4< / span >
< a id = "__codelineno-25-11" name = "__codelineno-25-11" href = "#__codelineno-25-11" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "na" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n5< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.rb< / span > < pre > < span > < / span > < code > < a id = "__codelineno-26-1" name = "__codelineno-26-1" href = "#__codelineno-26-1" > < / a >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.zig< / span > < pre > < span > < / span > < code > < a id = "__codelineno-27-1" name = "__codelineno-27-1" href = "#__codelineno-27-1" > < / a >
< / code > < / pre > < / div >
< / div >
< / div >
< / div >
< details class = "pythontutor" >
< summary > Code visualization< / summary >
< p > https://pythontutor.com/render.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%8F%89%E6%A0%91%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E8%8A%82%E7%82%B9%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%EF%BC%88%E6%8C%87%E9%92%88%EF%BC%89%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5& cumulative=false& curInstr=3& heapPrimitives=nevernest& mode=display& origin=opt-frontend.js& py=311& rawInputLstJSON=%5B%5D& textReferences=false< / p >
< / details >
< h3 id = "2-inserting-and-removing-nodes" > 2. Inserting and removing nodes< a class = "headerlink" href = "#2-inserting-and-removing-nodes" title = "Permanent link" > ¶ < / a > < / h3 >
< p > Similar to a linked list, inserting and removing nodes in a binary tree can be achieved by modifying pointers. The Figure 7-3 provides an example.< / p >
< p > < a class = "glightbox" href = "../binary_tree.assets/binary_tree_add_remove.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Inserting and removing nodes in a binary tree" class = "animation-figure" src = "../binary_tree.assets/binary_tree_add_remove.png" / > < / a > < / p >
< p align = "center" > Figure 7-3 Inserting and removing nodes in a binary tree < / p >
< div class = "tabbed-set tabbed-alternate" data-tabs = "3:14" > < input checked = "checked" id = "__tabbed_3_1" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_2" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_3" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_4" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_5" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_6" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_7" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_8" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_9" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_10" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_11" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_12" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_13" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_14" name = "__tabbed_3" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_3_1" > Python< / label > < label for = "__tabbed_3_2" > C++< / label > < label for = "__tabbed_3_3" > Java< / label > < label for = "__tabbed_3_4" > C#< / label > < label for = "__tabbed_3_5" > Go< / label > < label for = "__tabbed_3_6" > Swift< / label > < label for = "__tabbed_3_7" > JS< / label > < label for = "__tabbed_3_8" > TS< / label > < label for = "__tabbed_3_9" > Dart< / label > < label for = "__tabbed_3_10" > Rust< / label > < label for = "__tabbed_3_11" > C< / label > < label for = "__tabbed_3_12" > Kotlin< / label > < label for = "__tabbed_3_13" > Ruby< / label > < label for = "__tabbed_3_14" > Zig< / label > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-28-1" name = "__codelineno-28-1" href = "#__codelineno-28-1" > < / a > < span class = "c1" > # Inserting and removing nodes< / span >
< a id = "__codelineno-28-2" name = "__codelineno-28-2" href = "#__codelineno-28-2" > < / a > < span class = "n" > p< / span > < span class = "o" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 0< / span > < span class = "p" > )< / span >
< a id = "__codelineno-28-3" name = "__codelineno-28-3" href = "#__codelineno-28-3" > < / a > < span class = "c1" > # Inserting node P between n1 -> n2< / span >
< a id = "__codelineno-28-4" name = "__codelineno-28-4" href = "#__codelineno-28-4" > < / a > < span class = "n" > n1< / span > < span class = "o" > .< / span > < span class = "n" > left< / span > < span class = "o" > =< / span > < span class = "n" > p< / span >
< a id = "__codelineno-28-5" name = "__codelineno-28-5" href = "#__codelineno-28-5" > < / a > < span class = "n" > p< / span > < span class = "o" > .< / span > < span class = "n" > left< / span > < span class = "o" > =< / span > < span class = "n" > n2< / span >
< a id = "__codelineno-28-6" name = "__codelineno-28-6" href = "#__codelineno-28-6" > < / a > < span class = "c1" > # Removing node P< / span >
< a id = "__codelineno-28-7" name = "__codelineno-28-7" href = "#__codelineno-28-7" > < / a > < span class = "n" > n1< / span > < span class = "o" > .< / span > < span class = "n" > left< / span > < span class = "o" > =< / span > < span class = "n" > n2< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.cpp< / span > < pre > < span > < / span > < code > < a id = "__codelineno-29-1" name = "__codelineno-29-1" href = "#__codelineno-29-1" > < / a > < span class = "cm" > /* Inserting and removing nodes */< / span >
< a id = "__codelineno-29-2" name = "__codelineno-29-2" href = "#__codelineno-29-2" > < / a > < span class = "n" > TreeNode< / span > < span class = "o" > *< / span > < span class = "w" > < / span > < span class = "n" > P< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 0< / span > < span class = "p" > );< / span >
< a id = "__codelineno-29-3" name = "__codelineno-29-3" href = "#__codelineno-29-3" > < / a > < span class = "c1" > // Inserting node P between n1 and n2< / span >
< a id = "__codelineno-29-4" name = "__codelineno-29-4" href = "#__codelineno-29-4" > < / a > < span class = "n" > n1< / span > < span class = "o" > -> < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > P< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-29-5" name = "__codelineno-29-5" href = "#__codelineno-29-5" > < / a > < span class = "n" > P< / span > < span class = "o" > -> < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-29-6" name = "__codelineno-29-6" href = "#__codelineno-29-6" > < / a > < span class = "c1" > // Removing node P< / span >
< a id = "__codelineno-29-7" name = "__codelineno-29-7" href = "#__codelineno-29-7" > < / a > < span class = "n" > n1< / span > < span class = "o" > -> < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-30-1" name = "__codelineno-30-1" href = "#__codelineno-30-1" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > P< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 0< / span > < span class = "p" > );< / span >
< a id = "__codelineno-30-2" name = "__codelineno-30-2" href = "#__codelineno-30-2" > < / a > < span class = "c1" > // Inserting node P between n1 and n2< / span >
< a id = "__codelineno-30-3" name = "__codelineno-30-3" href = "#__codelineno-30-3" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "na" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > P< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-30-4" name = "__codelineno-30-4" href = "#__codelineno-30-4" > < / a > < span class = "n" > P< / span > < span class = "p" > .< / span > < span class = "na" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-30-5" name = "__codelineno-30-5" href = "#__codelineno-30-5" > < / a > < span class = "c1" > // Removing node P< / span >
< a id = "__codelineno-30-6" name = "__codelineno-30-6" href = "#__codelineno-30-6" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "na" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-31-1" name = "__codelineno-31-1" href = "#__codelineno-31-1" > < / a > < span class = "cm" > /* Inserting and removing nodes */< / span >
< a id = "__codelineno-31-2" name = "__codelineno-31-2" href = "#__codelineno-31-2" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > P< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "p" > (< / span > < span class = "m" > 0< / span > < span class = "p" > );< / span >
< a id = "__codelineno-31-3" name = "__codelineno-31-3" href = "#__codelineno-31-3" > < / a > < span class = "c1" > // Inserting node P between n1 and n2< / span >
< a id = "__codelineno-31-4" name = "__codelineno-31-4" href = "#__codelineno-31-4" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > P< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-31-5" name = "__codelineno-31-5" href = "#__codelineno-31-5" > < / a > < span class = "n" > P< / span > < span class = "p" > .< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-31-6" name = "__codelineno-31-6" href = "#__codelineno-31-6" > < / a > < span class = "c1" > // Removing node P< / span >
< a id = "__codelineno-31-7" name = "__codelineno-31-7" href = "#__codelineno-31-7" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-32-1" name = "__codelineno-32-1" href = "#__codelineno-32-1" > < / a > < span class = "cm" > /* Inserting and removing nodes */< / span >
< a id = "__codelineno-32-2" name = "__codelineno-32-2" href = "#__codelineno-32-2" > < / a > < span class = "c1" > // Inserting node P between n1 and n2< / span >
< a id = "__codelineno-32-3" name = "__codelineno-32-3" href = "#__codelineno-32-3" > < / a > < span class = "nx" > p< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nx" > NewTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 0< / span > < span class = "p" > )< / span >
< a id = "__codelineno-32-4" name = "__codelineno-32-4" href = "#__codelineno-32-4" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > Left< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > p< / span >
< a id = "__codelineno-32-5" name = "__codelineno-32-5" href = "#__codelineno-32-5" > < / a > < span class = "nx" > p< / span > < span class = "p" > .< / span > < span class = "nx" > Left< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span >
< a id = "__codelineno-32-6" name = "__codelineno-32-6" href = "#__codelineno-32-6" > < / a > < span class = "c1" > // Removing node P< / span >
< a id = "__codelineno-32-7" name = "__codelineno-32-7" href = "#__codelineno-32-7" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > Left< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.swift< / span > < pre > < span > < / span > < code > < a id = "__codelineno-33-1" name = "__codelineno-33-1" href = "#__codelineno-33-1" > < / a > < span class = "kd" > let< / span > < span class = "nv" > P< / span > < span class = "p" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > x< / span > < span class = "p" > :< / span > < span class = "mi" > 0< / span > < span class = "p" > )< / span >
< a id = "__codelineno-33-2" name = "__codelineno-33-2" href = "#__codelineno-33-2" > < / a > < span class = "c1" > // Inserting node P between n1 and n2< / span >
< a id = "__codelineno-33-3" name = "__codelineno-33-3" href = "#__codelineno-33-3" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "kr" > left< / span > < span class = "p" > =< / span > < span class = "n" > P< / span >
< a id = "__codelineno-33-4" name = "__codelineno-33-4" href = "#__codelineno-33-4" > < / a > < span class = "n" > P< / span > < span class = "p" > .< / span > < span class = "kr" > left< / span > < span class = "p" > =< / span > < span class = "n" > n2< / span >
< a id = "__codelineno-33-5" name = "__codelineno-33-5" href = "#__codelineno-33-5" > < / a > < span class = "c1" > // Removing node P< / span >
< a id = "__codelineno-33-6" name = "__codelineno-33-6" href = "#__codelineno-33-6" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "kr" > left< / span > < span class = "p" > =< / span > < span class = "n" > n2< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.js< / span > < pre > < span > < / span > < code > < a id = "__codelineno-34-1" name = "__codelineno-34-1" href = "#__codelineno-34-1" > < / a > < span class = "cm" > /* Inserting and removing nodes */< / span >
< a id = "__codelineno-34-2" name = "__codelineno-34-2" href = "#__codelineno-34-2" > < / a > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "nx" > P< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 0< / span > < span class = "p" > );< / span >
< a id = "__codelineno-34-3" name = "__codelineno-34-3" href = "#__codelineno-34-3" > < / a > < span class = "c1" > // Inserting node P between n1 and n2< / span >
< a id = "__codelineno-34-4" name = "__codelineno-34-4" href = "#__codelineno-34-4" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > P< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-34-5" name = "__codelineno-34-5" href = "#__codelineno-34-5" > < / a > < span class = "nx" > P< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-34-6" name = "__codelineno-34-6" href = "#__codelineno-34-6" > < / a > < span class = "c1" > // Removing node P< / span >
< a id = "__codelineno-34-7" name = "__codelineno-34-7" href = "#__codelineno-34-7" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.ts< / span > < pre > < span > < / span > < code > < a id = "__codelineno-35-1" name = "__codelineno-35-1" href = "#__codelineno-35-1" > < / a > < span class = "cm" > /* Inserting and removing nodes */< / span >
< a id = "__codelineno-35-2" name = "__codelineno-35-2" href = "#__codelineno-35-2" > < / a > < span class = "kd" > const< / span > < span class = "w" > < / span > < span class = "nx" > P< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 0< / span > < span class = "p" > );< / span >
< a id = "__codelineno-35-3" name = "__codelineno-35-3" href = "#__codelineno-35-3" > < / a > < span class = "c1" > // Inserting node P between n1 and n2< / span >
< a id = "__codelineno-35-4" name = "__codelineno-35-4" href = "#__codelineno-35-4" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > P< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-35-5" name = "__codelineno-35-5" href = "#__codelineno-35-5" > < / a > < span class = "nx" > P< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-35-6" name = "__codelineno-35-6" href = "#__codelineno-35-6" > < / a > < span class = "c1" > // Removing node P< / span >
< a id = "__codelineno-35-7" name = "__codelineno-35-7" href = "#__codelineno-35-7" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.dart< / span > < pre > < span > < / span > < code > < a id = "__codelineno-36-1" name = "__codelineno-36-1" href = "#__codelineno-36-1" > < / a > < span class = "cm" > /* Inserting and removing nodes */< / span >
< a id = "__codelineno-36-2" name = "__codelineno-36-2" href = "#__codelineno-36-2" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > P< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 0< / span > < span class = "p" > );< / span >
< a id = "__codelineno-36-3" name = "__codelineno-36-3" href = "#__codelineno-36-3" > < / a > < span class = "c1" > // Inserting node P between n1 and n2< / span >
< a id = "__codelineno-36-4" name = "__codelineno-36-4" href = "#__codelineno-36-4" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > P< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-36-5" name = "__codelineno-36-5" href = "#__codelineno-36-5" > < / a > < span class = "n" > P< / span > < span class = "p" > .< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-36-6" name = "__codelineno-36-6" href = "#__codelineno-36-6" > < / a > < span class = "c1" > // Removing node P< / span >
< a id = "__codelineno-36-7" name = "__codelineno-36-7" href = "#__codelineno-36-7" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.rs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-37-1" name = "__codelineno-37-1" href = "#__codelineno-37-1" > < / a > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "n" > p< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > ::< span class = "n" > new< / span > < span class = "p" > (< / span > < span class = "mi" > 0< / span > < span class = "p" > );< / span >
< a id = "__codelineno-37-2" name = "__codelineno-37-2" href = "#__codelineno-37-2" > < / a > < span class = "c1" > // Inserting node P between n1 and n2< / span >
< a id = "__codelineno-37-3" name = "__codelineno-37-3" href = "#__codelineno-37-3" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "n" > borrow_mut< / span > < span class = "p" > ().< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nb" > Some< / span > < span class = "p" > (< / span > < span class = "n" > p< / span > < span class = "p" > .< / span > < span class = "n" > clone< / span > < span class = "p" > ());< / span >
< a id = "__codelineno-37-4" name = "__codelineno-37-4" href = "#__codelineno-37-4" > < / a > < span class = "n" > p< / span > < span class = "p" > .< / span > < span class = "n" > borrow_mut< / span > < span class = "p" > ().< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nb" > Some< / span > < span class = "p" > (< / span > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "n" > clone< / span > < span class = "p" > ());< / span >
< a id = "__codelineno-37-5" name = "__codelineno-37-5" href = "#__codelineno-37-5" > < / a > < span class = "c1" > // Removing node P< / span >
< a id = "__codelineno-37-6" name = "__codelineno-37-6" href = "#__codelineno-37-6" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "n" > borrow_mut< / span > < span class = "p" > ().< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nb" > Some< / span > < span class = "p" > (< / span > < span class = "n" > n2< / span > < span class = "p" > );< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-38-1" name = "__codelineno-38-1" href = "#__codelineno-38-1" > < / a > < span class = "cm" > /* Inserting and removing nodes */< / span >
< a id = "__codelineno-38-2" name = "__codelineno-38-2" href = "#__codelineno-38-2" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "n" > P< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > newTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 0< / span > < span class = "p" > );< / span >
< a id = "__codelineno-38-3" name = "__codelineno-38-3" href = "#__codelineno-38-3" > < / a > < span class = "c1" > // Inserting node P between n1 and n2< / span >
< a id = "__codelineno-38-4" name = "__codelineno-38-4" href = "#__codelineno-38-4" > < / a > < span class = "n" > n1< / span > < span class = "o" > -> < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > P< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-38-5" name = "__codelineno-38-5" href = "#__codelineno-38-5" > < / a > < span class = "n" > P< / span > < span class = "o" > -> < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-38-6" name = "__codelineno-38-6" href = "#__codelineno-38-6" > < / a > < span class = "c1" > // Removing node P< / span >
< a id = "__codelineno-38-7" name = "__codelineno-38-7" href = "#__codelineno-38-7" > < / a > < span class = "n" > n1< / span > < span class = "o" > -> < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.kt< / span > < pre > < span > < / span > < code > < a id = "__codelineno-39-1" name = "__codelineno-39-1" href = "#__codelineno-39-1" > < / a > < span class = "kd" > val< / span > < span class = "w" > < / span > < span class = "nv" > P< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 0< / span > < span class = "p" > )< / span >
< a id = "__codelineno-39-2" name = "__codelineno-39-2" href = "#__codelineno-39-2" > < / a > < span class = "c1" > // Inserting node P between n1 and n2< / span >
< a id = "__codelineno-39-3" name = "__codelineno-39-3" href = "#__codelineno-39-3" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "na" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > P< / span >
< a id = "__codelineno-39-4" name = "__codelineno-39-4" href = "#__codelineno-39-4" > < / a > < span class = "n" > P< / span > < span class = "p" > .< / span > < span class = "na" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span >
< a id = "__codelineno-39-5" name = "__codelineno-39-5" href = "#__codelineno-39-5" > < / a > < span class = "c1" > // Removing node P< / span >
< a id = "__codelineno-39-6" name = "__codelineno-39-6" href = "#__codelineno-39-6" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "na" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.rb< / span > < pre > < span > < / span > < code > < a id = "__codelineno-40-1" name = "__codelineno-40-1" href = "#__codelineno-40-1" > < / a >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.zig< / span > < pre > < span > < / span > < code > < a id = "__codelineno-41-1" name = "__codelineno-41-1" href = "#__codelineno-41-1" > < / a >
< / code > < / pre > < / div >
< / div >
< / div >
< / div >
< details class = "pythontutor" >
< summary > Code visualization< / summary >
< p > https://pythontutor.com/render.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%8F%89%E6%A0%91%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E8%8A%82%E7%82%B9%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%EF%BC%88%E6%8C%87%E9%92%88%EF%BC%89%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5%0A%0A%20%20%20%20%23%20%E6%8F%92%E5%85%A5%E4%B8%8E%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%0A%20%20%20%20p%20%3D%20TreeNode%280%29%0A%20%20%20%20%23%20%E5%9C%A8%20n1%20-%3E%20n2%20%E4%B8%AD%E9%97%B4%E6%8F%92%E5%85%A5%E8%8A%82%E7%82%B9%20P%0A%20%20%20%20n1.left%20%3D%20p%0A%20%20%20%20p.left%20%3D%20n2%0A%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%20P%0A%20%20%20%20n1.left%20%3D%20n2& cumulative=false& curInstr=37& heapPrimitives=nevernest& mode=display& origin=opt-frontend.js& py=311& rawInputLstJSON=%5B%5D& textReferences=false< / p >
< / details >
< div class = "admonition note" >
< p class = "admonition-title" > Note< / p >
< p > It's important to note that inserting nodes may change the original logical structure of the binary tree, while removing nodes usually means removing the node and all its subtrees. Therefore, in a binary tree, insertion and removal are usually performed through a set of operations to achieve meaningful actions.< / p >
< / div >
< h2 id = "713-common-types-of-binary-trees" > 7.1.3 Common types of binary trees< a class = "headerlink" href = "#713-common-types-of-binary-trees" title = "Permanent link" > ¶ < / a > < / h2 >
< h3 id = "1-perfect-binary-tree" > 1. Perfect binary tree< a class = "headerlink" href = "#1-perfect-binary-tree" title = "Permanent link" > ¶ < / a > < / h3 >
< p > As shown in the Figure 7-4 , in a "perfect binary tree," all levels of nodes are fully filled. In a perfect binary tree, the degree of leaf nodes is < span class = "arithmatex" > \(0\)< / span > , and the degree of all other nodes is < span class = "arithmatex" > \(2\)< / span > ; if the tree's height is < span class = "arithmatex" > \(h\)< / span > , then the total number of nodes is < span class = "arithmatex" > \(2^{h+1} - 1\)< / span > , showing a standard exponential relationship, reflecting the common phenomenon of cell division in nature.< / p >
< div class = "admonition tip" >
< p class = "admonition-title" > Tip< / p >
< p > Please note that in the Chinese community, a perfect binary tree is often referred to as a "full binary tree."< / p >
< / div >
< p > < a class = "glightbox" href = "../binary_tree.assets/perfect_binary_tree.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Perfect binary tree" class = "animation-figure" src = "../binary_tree.assets/perfect_binary_tree.png" / > < / a > < / p >
< p align = "center" > Figure 7-4 Perfect binary tree < / p >
< h3 id = "2-complete-binary-tree" > 2. Complete binary tree< a class = "headerlink" href = "#2-complete-binary-tree" title = "Permanent link" > ¶ < / a > < / h3 >
< p > As shown in the Figure 7-5 , a "complete binary tree" has only the bottom level nodes not fully filled, and the bottom level nodes are filled as far left as possible.< / p >
< p > < a class = "glightbox" href = "../binary_tree.assets/complete_binary_tree.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Complete binary tree" class = "animation-figure" src = "../binary_tree.assets/complete_binary_tree.png" / > < / a > < / p >
< p align = "center" > Figure 7-5 Complete binary tree < / p >
< h3 id = "3-full-binary-tree" > 3. Full binary tree< a class = "headerlink" href = "#3-full-binary-tree" title = "Permanent link" > ¶ < / a > < / h3 >
< p > As shown in the Figure 7-6 , a "full binary tree" has all nodes except leaf nodes having two children.< / p >
< p > < a class = "glightbox" href = "../binary_tree.assets/full_binary_tree.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Full binary tree" class = "animation-figure" src = "../binary_tree.assets/full_binary_tree.png" / > < / a > < / p >
< p align = "center" > Figure 7-6 Full binary tree < / p >
< h3 id = "4-balanced-binary-tree" > 4. Balanced binary tree< a class = "headerlink" href = "#4-balanced-binary-tree" title = "Permanent link" > ¶ < / a > < / h3 >
< p > As shown in the Figure 7-7 , in a "balanced binary tree," the absolute difference in height between the left and right subtrees of any node does not exceed 1.< / p >
< p > < a class = "glightbox" href = "../binary_tree.assets/balanced_binary_tree.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "Balanced binary tree" class = "animation-figure" src = "../binary_tree.assets/balanced_binary_tree.png" / > < / a > < / p >
< p align = "center" > Figure 7-7 Balanced binary tree < / p >
< h2 id = "714-degeneration-of-binary-trees" > 7.1.4 Degeneration of binary trees< a class = "headerlink" href = "#714-degeneration-of-binary-trees" title = "Permanent link" > ¶ < / a > < / h2 >
< p > The Figure 7-8 shows the ideal and degenerate structures of binary trees. When every level of a binary tree is filled, it reaches the "perfect binary tree"; when all nodes are biased towards one side, the binary tree degenerates into a "linked list".< / p >
< ul >
< li > The perfect binary tree is the ideal situation, fully leveraging the "divide and conquer" advantage of binary trees.< / li >
< li > A linked list is another extreme, where operations become linear, degrading the time complexity to < span class = "arithmatex" > \(O(n)\)< / span > .< / li >
< / ul >
< p > < a class = "glightbox" href = "../binary_tree.assets/binary_tree_best_worst_cases.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "The Best and Worst Structures of Binary Trees" class = "animation-figure" src = "../binary_tree.assets/binary_tree_best_worst_cases.png" / > < / a > < / p >
< p align = "center" > Figure 7-8 The Best and Worst Structures of Binary Trees < / p >
< p > As shown in the Table 7-1 , in the best and worst structures, the number of leaf nodes, total number of nodes, and height of the binary tree reach their maximum or minimum values.< / p >
< p align = "center" > Table 7-1 The Best and Worst Structures of Binary Trees < / p >
< div class = "center-table" >
< table >
< thead >
< tr >
< th > < / th >
< th > Perfect binary tree< / th >
< th > Linked list< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td > Number of nodes at level < span class = "arithmatex" > \(i\)< / span > < / td >
< td > < span class = "arithmatex" > \(2^{i-1}\)< / span > < / td >
< td > < span class = "arithmatex" > \(1\)< / span > < / td >
< / tr >
< tr >
< td > Number of leaf nodes in a tree with height < span class = "arithmatex" > \(h\)< / span > < / td >
< td > < span class = "arithmatex" > \(2^h\)< / span > < / td >
< td > < span class = "arithmatex" > \(1\)< / span > < / td >
< / tr >
< tr >
< td > Total number of nodes in a tree with height < span class = "arithmatex" > \(h\)< / span > < / td >
< td > < span class = "arithmatex" > \(2^{h+1} - 1\)< / span > < / td >
< td > < span class = "arithmatex" > \(h + 1\)< / span > < / td >
< / tr >
< tr >
< td > Height of a tree with < span class = "arithmatex" > \(n\)< / span > total nodes< / td >
< td > < span class = "arithmatex" > \(\log_2 (n+1) - 1\)< / span > < / td >
< td > < span class = "arithmatex" > \(n - 1\)< / span > < / td >
< / tr >
< / tbody >
< / table >
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< script > document $ . subscribe ( ( ) => { const lightbox = GLightbox ( { "touchNavigation" : true , "loop" : false , "zoomable" : true , "draggable" : false , "openEffect" : "zoom" , "closeEffect" : "zoom" , "slideEffect" : "none" } ) ; } ) < / script > < / body >
< / html >
