@@ -1435,6 +1435,26 @@
7.2.1 层序遍历
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1. 代码实现
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2. 复杂度分析
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@@ -1442,6 +1462,26 @@
7.2.2 前序、中序、后序遍历
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< nav class = "md-nav" aria-label = "7.2.2 >
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1. 代码实现
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2. 复杂度分析
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@@ -3409,6 +3449,26 @@
7.2.1 层序遍历
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< nav class = "md-nav" aria-label = "7.2.1 >
< ul class = "md-nav__list" >
< li class = "md-nav__item" >
< a href = "#1" class = "md-nav__link" >
1. 代码实现
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< li class = "md-nav__item" >
< a href = "#2" class = "md-nav__link" >
2. 复杂度分析
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@@ -3416,6 +3476,26 @@
7.2.2 前序、中序、后序遍历
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< nav class = "md-nav" aria-label = "7.2.2 >
< ul class = "md-nav__list" >
< li class = "md-nav__item" >
< a href = "#1_1" class = "md-nav__link" >
1. 代码实现
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< li class = "md-nav__item" >
< a href = "#2_1" class = "md-nav__link" >
2. 复杂度分析
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@@ -3450,6 +3530,7 @@
< p > < img alt = "二叉树的层序遍历" src = "../binary_tree_traversal.assets/binary_tree_bfs.png" / > < / p >
< p align = "center" > 图 7-9 二叉树的层序遍历 < / p >
< h3 id = "1" > 1. 代码实现< a class = "headerlink" href = "#1" title = "Permanent link" > ¶ < / a > < / h3 >
< p > 广度优先遍历通常借助“队列”来实现。队列遵循“先进先出”的规则，而广度优先遍历则遵循“逐层推进”的规则，两者背后的思想是一致的。< / p >
< div class = "tabbed-set tabbed-alternate" data-tabs = "1:12" > < 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" / > < div class = "tabbed-labels" > < label for = "__tabbed_1_1" > Java< / label > < label for = "__tabbed_1_2" > C++< / label > < label for = "__tabbed_1_3" > Python< / label > < label for = "__tabbed_1_4" > Go< / label > < label for = "__tabbed_1_5" > JS< / label > < label for = "__tabbed_1_6" > TS< / label > < label for = "__tabbed_1_7" > C< / label > < label for = "__tabbed_1_8" > C#< / label > < label for = "__tabbed_1_9" > Swift< / label > < label for = "__tabbed_1_10" > Zig< / label > < label for = "__tabbed_1_11" > Dart< / label > < label for = "__tabbed_1_12" > Rust< / label > < / div >
< div class = "tabbed-content" >
@@ -3733,14 +3814,19 @@
< / div >
< / div >
< / div >
< p > < strong > 时间< / strong > ：所有节点被访问一次，使用 < span class = "arithmatex" > \(O(n)\)< / span > 时间，其中 < span class = "arithmatex" > \(n\)< / span > 为节点数量。< / p >
< p > < strong > 空间复杂度< / strong > ：在最差情况下，即满二叉树时，遍历到最底层之前，队列中最多同时存在 < span class = "arithmatex" > \((n + 1) / 2\)< / span > 个节点，占用 < span class = "arithmatex" > \(O(n)\)< / span > 空间。< / p >
< h3 id = "2" > 2. 分析 < a class = "headerlink" href = "#2" title = "Permanent link" > ¶ < / a > < / h3 >
< ul >
< li > < strong > 时间复杂度 < span class = "arithmatex" > \(O(n)\)< / span > < / strong > ：所有节点被访问一次，使用 < span class = "arithmatex" > \(O(n)\)< / span > 时间，其中 < span class = "arithmatex" > \(n\)< / span > 为节点数量。< / li >
< li > < strong > 空间复杂度 < span class = "arithmatex" > \(O(n)\)< / span > < / strong > ：在最差情况下，即满二叉树时，遍历到最底层之前，队列中最多同时存在 < span class = "arithmatex" > \((n + 1) / 2\)< / span > 个节点，占用 < span class = "arithmatex" > \(O(n)\)< / span > 空间。< / li >
< / ul >
< h2 id = "722" > 7.2.2 前序、中序、后序遍历< a class = "headerlink" href = "#722" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 相应地，前序、中序和后序遍历都属于「深度优先遍历 depth-first traversal」， ， < / p >
< p > 图 7-10 展示了对二叉树进行深度优先遍历的工作原理。< strong > 深度优先遍历就像是绕着整个二叉树的外围“走”一圈< / strong > ，在每个节点都会遇到三个位置，分别对应前序遍历、中序遍历和后序遍历。< / p >
< p > < img alt = "二叉搜索树的前、中、后序遍历" src = "../binary_tree_traversal.assets/binary_tree_dfs.png" / > < / p >
< p align = "center" > 图 7-10 二叉搜索树的前、中、后序遍历 < / p >
< h3 id = "1_1" > 1. 代码实现< a class = "headerlink" href = "#1_1" title = "Permanent link" > ¶ < / a > < / h3 >
< p > 深度优先搜索通常基于递归实现：< / p >
< div class = "tabbed-set tabbed-alternate" data-tabs = "2:12" > < 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" / > < div class = "tabbed-labels" > < label for = "__tabbed_2_1" > Java< / label > < label for = "__tabbed_2_2" > C++< / label > < label for = "__tabbed_2_3" > Python< / label > < label for = "__tabbed_2_4" > Go< / label > < label for = "__tabbed_2_5" > JS< / label > < label for = "__tabbed_2_6" > TS< / label > < label for = "__tabbed_2_7" > C< / label > < label for = "__tabbed_2_8" > C#< / label > < label for = "__tabbed_2_9" > Swift< / label > < label for = "__tabbed_2_10" > Zig< / label > < label for = "__tabbed_2_11" > Dart< / label > < label for = "__tabbed_2_12" > Rust< / label > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
@@ -4132,11 +4218,9 @@
< / div >
< / div >
< / div >
< p > < strong > 时间复杂度< / strong > ：所有节点被访问一次，使用 < span class = "arithmatex" > \(O(n)\)< / span > 时间，其中 < span class = "arithmatex" > \(n\)< / span > 为节点数量。< / p >
< p > < strong > 空间复杂度< / strong > ：在最差情况下，即树退化为链表时，递归深度达到 < span class = "arithmatex" > \(n\)< / span > ，系统占用 < span class = "arithmatex" > \(O(n)\)< / span > 栈帧空间。< / p >
< div class = "admonition note" >
< p class = "admonition-title" > Note< / p >
< p > 我们也可以不使用递归，仅基于迭代实现前、中、后序遍历 ，有兴趣的同学可以自行实现 。< / p >
< p > 深度优先搜索也可以基于迭代实现 ，有兴趣的同学可以自行研究 。< / p >
< / div >
< p > 图 7-11 展示了前序遍历二叉树的递归过程，其可分为“递”和“归”两个逆向的部分。< / p >
< ol >
@@ -4182,6 +4266,12 @@
< / div >
< p align = "center" > 图 7-11 前序遍历的递归过程 < / p >
< h3 id = "2_1" > 2. 复杂度分析< a class = "headerlink" href = "#2_1" title = "Permanent link" > ¶ < / a > < / h3 >
< ul >
< li > < strong > 时间复杂度 < span class = "arithmatex" > \(O(n)\)< / span > < / strong > ：所有节点被访问一次，使用 < span class = "arithmatex" > \(O(n)\)< / span > 时间。< / li >
< li > < strong > 空间复杂度 < span class = "arithmatex" > \(O(n)\)< / span > < / strong > ：在最差情况下，即树退化为链表时，递归深度达到 < span class = "arithmatex" > \(n\)< / span > ，系统占用 < span class = "arithmatex" > \(O(n)\)< / span > 栈帧空间。< / li >
< / ul >
krahets