From b6b03ef84bef28fe96912ddcd83e5858932c23d0 Mon Sep 17 00:00:00 2001 From: krahets Date: Sat, 17 Jun 2023 00:45:41 +0800 Subject: [PATCH] Fix the figures of binary tree. Replace null with None. --- .../linkedlist_common_types.png | Bin 65337 -> 67092 bytes .../linkedlist_definition.png | Bin 78698 -> 79069 bytes .../linked_list.md | 4 ++-- docs/chapter_preface/suggestions.md | 15 ++++++--------- .../array_representation_binary_tree.png | Bin 0 -> 87846 bytes ...ay_representation_complete_binary_tree.png | Bin 0 -> 69038 bytes .../array_representation_with_empty.png | Bin 0 -> 76877 bytes .../array_representation_without_empty.png | Bin 0 -> 67730 bytes .../array_representation_of_tree.md | 16 ++++++++-------- .../binary_search_tree.assets/bst_insert.png | Bin 87902 -> 88287 bytes .../bst_remove_case1.png | Bin 87327 -> 87796 bytes docs/chapter_tree/binary_search_tree.md | 6 +++--- ...ay_representation_complete_binary_tree.png | Bin 72514 -> 0 bytes .../array_representation_mapping.png | Bin 93094 -> 0 bytes .../array_representation_with_empty.png | Bin 76750 -> 0 bytes .../array_representation_without_empty.png | Bin 67561 -> 0 bytes .../balanced_binary_tree.png | Bin 58878 -> 59118 bytes .../binary_tree_best_worst_cases.png | Bin 0 -> 64155 bytes .../binary_tree_corner_cases.png | Bin 63615 -> 0 bytes .../complete_binary_tree.png | Bin 48616 -> 53442 bytes .../binary_tree.assets/full_binary_tree.png | Bin 43107 -> 45264 bytes .../perfect_binary_tree.png | Bin 57892 -> 60664 bytes docs/chapter_tree/binary_tree.md | 4 ++-- 23 files changed, 21 insertions(+), 24 deletions(-) create mode 100644 docs/chapter_tree/array_representation_of_tree.assets/array_representation_binary_tree.png create mode 100644 docs/chapter_tree/array_representation_of_tree.assets/array_representation_complete_binary_tree.png create mode 100644 docs/chapter_tree/array_representation_of_tree.assets/array_representation_with_empty.png create mode 100644 docs/chapter_tree/array_representation_of_tree.assets/array_representation_without_empty.png delete mode 100644 docs/chapter_tree/binary_tree.assets/array_representation_complete_binary_tree.png 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a/docs/chapter_tree/array_representation_of_tree.md b/docs/chapter_tree/array_representation_of_tree.md index 8d4cf2727..c4e577eaa 100644 --- a/docs/chapter_tree/array_representation_of_tree.md +++ b/docs/chapter_tree/array_representation_of_tree.md @@ -10,17 +10,17 @@ 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”:**若节点的索引为 $i$ ,则该节点的左子节点索引为 $2i + 1$ ,右子节点索引为 $2i + 2$** 。 -![完美二叉树的数组表示](binary_tree.assets/array_representation_mapping.png) +![完美二叉树的数组表示](array_representation_of_tree.assets/array_representation_binary_tree.png) **映射公式的作用相当于链表中的指针**。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。 ## 表示任意二叉树 -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 $\text{null}$ ,而层序遍历序列并不包含这些 $\text{null}$ 。我们无法仅凭该序列来推测 $\text{null}$ 的数量和分布位置,**这意味着存在多种二叉树结构都符合该层序遍历序列**。显然在这种情况下,上述的数组表示方法已经失效。 +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 $\text{None}$ ,而层序遍历序列并不包含这些 $\text{None}$ 。我们无法仅凭该序列来推测 $\text{None}$ 的数量和分布位置,**这意味着存在多种二叉树结构都符合该层序遍历序列**。显然在这种情况下,上述的数组表示方法已经失效。 -![层序遍历序列对应多种二叉树可能性](binary_tree.assets/array_representation_without_empty.png) +![层序遍历序列对应多种二叉树可能性](array_representation_of_tree.assets/array_representation_without_empty.png) -为了解决此问题,**我们可以考虑在层序遍历序列中显式地写出所有 $\text{null}$**。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。 +为了解决此问题,**我们可以考虑在层序遍历序列中显式地写出所有 $\text{None}$**。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。 === "Java" @@ -108,7 +108,7 @@ List tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15]; ``` -![任意类型二叉树的数组表示](binary_tree.assets/array_representation_with_empty.png) +![任意类型二叉树的数组表示](array_representation_of_tree.assets/array_representation_with_empty.png) ## 优势与局限性 @@ -122,8 +122,8 @@ - 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。 - 增删节点需要通过数组插入与删除操作实现,效率较低; -- 当二叉树中存在大量 $\text{null}$ 时,数组中包含的节点数据比重较低,空间利用率较低。 +- 当二叉树中存在大量 $\text{None}$ 时,数组中包含的节点数据比重较低,空间利用率较低。 -**完全二叉树非常适合使用数组来表示**。回顾完全二叉树的定义,$\text{null}$ 只出现在最底层且靠右的位置,**这意味着所有 $\text{null}$ 一定出现在层序遍历序列的末尾**。因此,在使用数组表示完全二叉树时,可以省略存储所有 $\text{null}$ 。 +**完全二叉树非常适合使用数组来表示**。回顾完全二叉树的定义,$\text{None}$ 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