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3. 数据结构
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3.2. 基本数据类型
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3.3. 数字编码 *
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3.4. 字符编码 *
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3.5. 小结
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4. 数组与链表
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4.1. 数组
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4.2. 链表
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4.3. 列表
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4.4. 小结
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5. 栈与队列
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5.1. 栈
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5.2. 队列
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5.3. 双向队列
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5.4. 小结
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6. 散列表
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6.1. 哈希表（ ）
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6.2. 哈希冲突（ ）
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6.3. 哈希算法（ ）
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6.4. 小结
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7. 树
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< a href = "../binary_tree/" class = "md-nav__link" >
7.1. 二叉树
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< a href = "../binary_tree_traversal/" class = "md-nav__link" >
7.2. 二叉树遍历
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7.3. 二叉树数组表示
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7.3. 二叉树数组表示
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目录
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7.3.1. 表示完美二叉树
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7.3.2. 表示任意二叉树
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7.3.3. 优势与局限性
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7.4. 二叉搜索树
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7.5. AVL 树 *
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< a href = "../summary/" class = "md-nav__link" >
7.6. 小结
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8. 堆
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8.1. 堆
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< a href = "../../chapter_heap/build_heap/" class = "md-nav__link" >
8.2. 建堆操作
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8.3. Top-K 问题（ ）
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8.4. 小结
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9.1. 图
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< a href = "../../chapter_graph/graph_operations/" class = "md-nav__link" >
9.2. 图基础操作
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9.3. 图的遍历
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9.4. 小结
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10. 搜索
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10.1. 二分查找
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10.2. 二分查找边界
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10.3. 哈希优化策略
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10.4. 重识搜索算法
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10.5. 小结
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11. 排序
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11.1. 排序算法
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11.2. 选择排序
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11.3. 冒泡排序
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11.4. 插入排序
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11.5. 快速排序
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11.6. 归并排序
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11.7. 堆排序
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11.8. 桶排序
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11.9. 计数排序
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11.10. 基数排序
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11.11. 小结
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13.1. 回溯算法
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13.2. 全排列问题
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13.3. 子集和问题
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13.4. N 皇后问题
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13.5. 小结
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14.1. 初探动态规划（ ）
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14.2. DP 问题特性（ ）
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14.3. DP 解题思路（ ）
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14.4. 0-1 背包问题（ ）
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14.5. 完全背包问题（ ）
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14.6. 编辑距离问题（ ）
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14.7. 小结（ ）
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15. 附录
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15.1. 编程环境安装
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15.2. 一起参与创作
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7.3.1. 表示完美二叉树
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7.3.2. 表示任意二叉树
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7.3.3. 优势与局限性
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M10 20H6V4h7v5h5v3.1l2-2V8l-6-6H6c-1.1 0-2 .9-2 2v16c0 1.1.9 2 2 2h4v-2m10.2-7c.1 0 .3.1.4.2l1.3 1.3c.2.2.2.6 0 .8l-1 1-2.1-2.1 1-1c.1-.1.2-.2.4-.2m0 3.9L14.1 23H12v-2.1l6.1-6.1 2.1 2.1Z" / > < / svg >
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< h1 id = "73" > 7.3. 二叉树数组表示< a class = "headerlink" href = "#73" title = "Permanent link" > ¶ < / a > < / h1 >
< p > 在链表表示下，二叉树的存储单元为节点 < code > TreeNode< / code > ，节点之间通过指针相连接。在上节中，我们学习了在链表表示下的二叉树的各项基本操作。< / p >
< p > 那么，能否用「数组」来表示二叉树呢？答案是肯定的。< / p >
< h2 id = "731" > 7.3.1. 表示完美二叉树< a class = "headerlink" href = "#731" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 先分析一个简单案例，给定一个完美二叉树，我们将节点按照层序遍历的顺序编号（从 < span class = "arithmatex" > \(0\)< / span > 开始），此时每个节点都对应唯一的索引。< / p >
< p > 根据层序遍历的特性，我们可以推导出父节点索引与子节点索引之间的“映射公式”：< strong > 若节点的索引为 < span class = "arithmatex" > \(i\)< / span > ，则该节点的左子节点索引为 < span class = "arithmatex" > \(2i + 1\)< / span > ，右子节点索引为 < span class = "arithmatex" > \(2i + 2\)< / span > < / strong > 。< / p >
< p > < img alt = "完美二叉树的数组表示" src = "../array_representation_of_tree.assets/array_representation_binary_tree.png" / > < / p >
< p align = "center" > Fig. 完美二叉树的数组表示 < / p >
< p > < strong > 映射公式的作用相当于链表中的指针< / strong > 。如果我们将节点按照层序遍历的顺序存储在一个数组中，那么对于数组中的任意节点，我们都可以通过映射公式来访问其子节点。< / p >
< h2 id = "732" > 7.3.2. 表示任意二叉树< a class = "headerlink" href = "#732" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 然而，完美二叉树只是一个特例。在二叉树的中间层，通常存在许多 < span class = "arithmatex" > \(\text{None}\)< / span > ，而层序遍历序列并不包含这些 < span class = "arithmatex" > \(\text{None}\)< / span > 。我们无法仅凭该序列来推测 < span class = "arithmatex" > \(\text{None}\)< / span > 的数量和分布位置，< strong > 这意味着存在多种二叉树结构都符合该层序遍历序列< / strong > 。显然在这种情况下，上述的数组表示方法已经失效。< / p >
< p > < img alt = "层序遍历序列对应多种二叉树可能性" src = "../array_representation_of_tree.assets/array_representation_without_empty.png" / > < / p >
< p align = "center" > Fig. 层序遍历序列对应多种二叉树可能性 < / p >
< p > 为了解决此问题，< strong > 我们可以考虑在层序遍历序列中显式地写出所有 < span class = "arithmatex" > \(\text{None}\)< / span > < / strong > 。如下图所示，这样处理后，层序遍历序列就可以唯一表示二叉树了。< / p >
< div class = "tabbed-set tabbed-alternate" data-tabs = "1:11" > < 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" / > < 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" > JavaScript< / label > < label for = "__tabbed_1_6" > TypeScript< / 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 > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "cm" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "c1" > // 使用 int 的包装类 Integer ，就可以使用 null 来标记空位< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "n" > Integer< / span > < span class = "o" > []< / span > < span class = "w" > < / span > < span class = "n" > tree< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > {< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 4< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 6< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 7< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 8< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 9< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 12< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 15< / span > < span class = "w" > < / span > < span class = "p" > };< / 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" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-1-2" name = "__codelineno-1-2" href = "#__codelineno-1-2" > < / a > < span class = "c1" > // 使用 int 最大值标记空位，因此要求节点值不能为 INT_MAX< / span >
< a id = "__codelineno-1-3" name = "__codelineno-1-3" href = "#__codelineno-1-3" > < / a > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > < / span > < span class = "w" > < / span > < span class = "n" > tree< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > {< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 4< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 6< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 7< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 8< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 9< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 12< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 15< / span > < span class = "p" > };< / span >
< / code > < / pre > < / div >
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< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-2-1" name = "__codelineno-2-1" href = "#__codelineno-2-1" > < / a > < span class = "c1" > # 二叉树的数组表示< / span >
< a id = "__codelineno-2-2" name = "__codelineno-2-2" href = "#__codelineno-2-2" > < / a > < span class = "c1" > # 直接使用 None 来表示空位< / span >
< a id = "__codelineno-2-3" name = "__codelineno-2-3" href = "#__codelineno-2-3" > < / a > < span class = "n" > tree< / span > < span class = "o" > =< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "mi" > 2< / span > < span class = "p" > ,< / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "mi" > 4< / span > < span class = "p" > ,< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span > < span class = "mi" > 6< / span > < span class = "p" > ,< / span > < span class = "mi" > 7< / span > < span class = "p" > ,< / span > < span class = "mi" > 8< / span > < span class = "p" > ,< / span > < span class = "mi" > 9< / span > < span class = "p" > ,< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span > < span class = "mi" > 12< / span > < span class = "p" > ,< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span > < span class = "mi" > 15< / span > < 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" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-3-2" name = "__codelineno-3-2" href = "#__codelineno-3-2" > < / a > < span class = "c1" > // 使用 any 类型的切片, 就可以使用 nil 来标记空位< / span >
< a id = "__codelineno-3-3" name = "__codelineno-3-3" href = "#__codelineno-3-3" > < / a > < span class = "nx" > tree< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "p" > []< / span > < span class = "kt" > any< / span > < span class = "p" > {< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 4< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 6< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 7< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 8< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 9< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 12< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 15< / span > < 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" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-4-2" name = "__codelineno-4-2" href = "#__codelineno-4-2" > < / a > < span class = "c1" > // 直接使用 null 来表示空位< / span >
< a id = "__codelineno-4-3" name = "__codelineno-4-3" href = "#__codelineno-4-3" > < / a > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "nx" > tree< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > [< / span > < span class = "mf" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 3< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 4< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 6< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 7< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 8< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 9< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 12< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 15< / span > < 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" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-5-2" name = "__codelineno-5-2" href = "#__codelineno-5-2" > < / a > < span class = "c1" > // 直接使用 null 来表示空位< / span >
< a id = "__codelineno-5-3" name = "__codelineno-5-3" href = "#__codelineno-5-3" > < / a > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "nx" > tree< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > number< / 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 = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > [< / span > < span class = "mf" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 3< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 4< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 6< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 7< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 8< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 9< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 12< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 15< / span > < 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" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-6-2" name = "__codelineno-6-2" href = "#__codelineno-6-2" > < / a > < span class = "c1" > // 使用 int 最大值标记空位，因此要求节点值不能为 INT_MAX< / span >
< a id = "__codelineno-6-3" name = "__codelineno-6-3" href = "#__codelineno-6-3" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > tree< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > {< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 4< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 6< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 7< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 8< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 9< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 12< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 15< / span > < 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" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-7-2" name = "__codelineno-7-2" href = "#__codelineno-7-2" > < / a > < span class = "c1" > // 使用 int? 可空类型 ，就可以使用 null 来标记空位< / span >
< a id = "__codelineno-7-3" name = "__codelineno-7-3" href = "#__codelineno-7-3" > < / a > < span class = "kt" > int?< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "n" > tree< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > {< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 3< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 4< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "k" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 6< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 7< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 8< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 9< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "k" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "k" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 12< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "k" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "k" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 15< / span > < span class = "w" > < / span > < 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" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-8-2" name = "__codelineno-8-2" href = "#__codelineno-8-2" > < / a > < span class = "c1" > // 使用 Int? 可空类型 ，就可以使用 nil 来标记空位< / span >
< a id = "__codelineno-8-3" name = "__codelineno-8-3" href = "#__codelineno-8-3" > < / a > < span class = "kd" > let< / span > < span class = "nv" > tree< / span > < span class = "p" > :< / span > < span class = "p" > [< / span > < span class = "nb" > Int< / span > < span class = "p" > ?]< / span > < span class = "p" > =< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "mi" > 2< / span > < span class = "p" > ,< / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "mi" > 4< / span > < span class = "p" > ,< / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "mi" > 6< / span > < span class = "p" > ,< / span > < span class = "mi" > 7< / span > < span class = "p" > ,< / span > < span class = "mi" > 8< / span > < span class = "p" > ,< / span > < span class = "mi" > 9< / span > < span class = "p" > ,< / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "mi" > 12< / span > < span class = "p" > ,< / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "mi" > 15< / span > < 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 >
< / 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" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-10-2" name = "__codelineno-10-2" href = "#__codelineno-10-2" > < / a > < span class = "c1" > // 使用 int? 可空类型 ，就可以使用 null 来标记空位< / span >
< a id = "__codelineno-10-3" name = "__codelineno-10-3" href = "#__codelineno-10-3" > < / a > < span class = "n" > List< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > ?> < / span > < span class = "w" > < / span > < span class = "n" > tree< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > [< / span > < span class = "m" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 3< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 4< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 6< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 7< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 8< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 9< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 12< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 15< / span > < span class = "p" > ];< / span >
< / code > < / pre > < / div >
< / div >
< / div >
< / div >
< p > < img alt = "任意类型二叉树的数组表示" src = "../array_representation_of_tree.assets/array_representation_with_empty.png" / > < / p >
< p align = "center" > Fig. 任意类型二叉树的数组表示 < / p >
< h2 id = "733" > 7.3.3. 优势与局限性< a class = "headerlink" href = "#733" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 二叉树的数组表示存在以下优点：< / p >
< ul >
< li > 数组存储在连续的内存空间中，缓存友好，访问与遍历速度较快；< / li >
< li > 不需要存储指针，比较节省空间；< / li >
< li > 允许随机访问节点；< / li >
< / ul >
< p > 然而，数组表示也具有一些局限性：< / p >
< ul >
< li > 数组存储需要连续内存空间，因此不适合存储数据量过大的树。< / li >
< li > 增删节点需要通过数组插入与删除操作实现，效率较低；< / li >
< li > 当二叉树中存在大量 < span class = "arithmatex" > \(\text{None}\)< / span > 时，数组中包含的节点数据比重较低，空间利用率较低。< / li >
< / ul >
< p > < strong > 完全二叉树非常适合使用数组来表示< / strong > 。回顾完全二叉树的定义，< span class = "arithmatex" > \(\text{None}\)< / span > 只出现在最底层且靠右的位置，< strong > 这意味着所有 < span class = "arithmatex" > \(\text{None}\)< / span > 一定出现在层序遍历序列的末尾< / strong > 。因此，在使用数组表示完全二叉树时，可以省略存储所有 < span class = "arithmatex" > \(\text{None}\)< / span > 。< / p >
< p > < img alt = "完全二叉树的数组表示" src = "../array_representation_of_tree.assets/array_representation_complete_binary_tree.png" / > < / p >
< p align = "center" > Fig. 完全二叉树的数组表示 < / p >
< h2 id = "__comments" > 评论< / h2 >
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