From 919236f479f8cae6fdbab53cb1e276578e11aee1 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Fri, 23 Dec 2022 00:44:40 +0800 Subject: [PATCH] 1. Add array representation of binary tree. 2. Update the LEGO example. 3. Update headers of the Python files. --- .../chapter_array_and_linkedlist/array.py | 4 +- .../linked_list.py | 4 +- .../chapter_array_and_linkedlist/list.py | 4 +- .../chapter_array_and_linkedlist/my_list.py | 4 +- .../leetcode_two_sum.py | 4 +- .../space_complexity.py | 4 +- .../time_complexity.py | 4 +- .../worst_best_time_complexity.py | 4 +- .../python/chapter_searching/binary_search.py | 4 +- .../chapter_searching/hashing_search.py | 4 +- .../python/chapter_searching/linear_search.py | 4 +- codes/python/chapter_sorting/bubble_sort.py | 4 +- .../python/chapter_sorting/insertion_sort.py | 4 +- codes/python/chapter_sorting/merge_sort.py | 4 +- codes/python/chapter_sorting/quick_sort.py | 4 +- .../chapter_stack_and_queue/array_queue.py | 4 +- .../chapter_stack_and_queue/array_stack.py | 4 +- codes/python/chapter_stack_and_queue/deque.py | 4 +- .../linkedlist_queue.py | 4 +- .../linkedlist_stack.py | 4 +- codes/python/chapter_stack_and_queue/queue.py | 4 +- codes/python/chapter_stack_and_queue/stack.py | 4 +- .../python/chapter_tree/binary_search_tree.py | 4 +- codes/python/chapter_tree/binary_tree.py | 4 +- codes/python/chapter_tree/binary_tree_bfs.py | 4 +- codes/python/chapter_tree/binary_tree_dfs.py | 4 +- codes/python/include/binary_tree.py | 4 +- codes/python/include/linked_list.py | 4 +- codes/python/include/print_util.py | 4 +- .../time_complexity.md | 2 +- docs/chapter_introduction/what_is_dsa.md | 15 +++- ...ay_representation_complete_binary_tree.png | Bin 0 -> 73985 bytes .../array_representation_mapping.png | Bin 0 -> 96663 bytes .../array_representation_with_empty.png | Bin 0 -> 78734 bytes .../array_representation_without_empty.png | Bin 0 -> 70009 bytes docs/chapter_tree/binary_tree.md | 79 ++++++++++++++++++ 36 files changed, 151 insertions(+), 61 deletions(-) create mode 100644 docs/chapter_tree/binary_tree.assets/array_representation_complete_binary_tree.png create mode 100644 docs/chapter_tree/binary_tree.assets/array_representation_mapping.png create mode 100644 docs/chapter_tree/binary_tree.assets/array_representation_with_empty.png create mode 100644 docs/chapter_tree/binary_tree.assets/array_representation_without_empty.png diff --git a/codes/python/chapter_array_and_linkedlist/array.py b/codes/python/chapter_array_and_linkedlist/array.py index 7f34472c7..8933fb482 100644 --- a/codes/python/chapter_array_and_linkedlist/array.py +++ b/codes/python/chapter_array_and_linkedlist/array.py @@ -1,8 +1,8 @@ -''' +""" File: array.py Created Time: 2022-11-25 Author: Krahets (krahets@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_array_and_linkedlist/linked_list.py b/codes/python/chapter_array_and_linkedlist/linked_list.py index d176d8dd2..dce110340 100644 --- a/codes/python/chapter_array_and_linkedlist/linked_list.py +++ b/codes/python/chapter_array_and_linkedlist/linked_list.py @@ -1,8 +1,8 @@ -''' +""" File: linked_list.py Created Time: 2022-11-25 Author: Krahets (krahets@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_array_and_linkedlist/list.py b/codes/python/chapter_array_and_linkedlist/list.py index ac71b98b8..b0e2f2536 100644 --- a/codes/python/chapter_array_and_linkedlist/list.py +++ b/codes/python/chapter_array_and_linkedlist/list.py @@ -1,8 +1,8 @@ -''' +""" File: list.py Created Time: 2022-11-25 Author: Krahets (krahets@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_array_and_linkedlist/my_list.py b/codes/python/chapter_array_and_linkedlist/my_list.py index 077fc0ce5..550e66955 100644 --- a/codes/python/chapter_array_and_linkedlist/my_list.py +++ b/codes/python/chapter_array_and_linkedlist/my_list.py @@ -1,8 +1,8 @@ -''' +""" File: my_list.py Created Time: 2022-11-25 Author: Krahets (krahets@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_computational_complexity/leetcode_two_sum.py b/codes/python/chapter_computational_complexity/leetcode_two_sum.py index 182caec39..cb3bf3b02 100644 --- a/codes/python/chapter_computational_complexity/leetcode_two_sum.py +++ b/codes/python/chapter_computational_complexity/leetcode_two_sum.py @@ -1,8 +1,8 @@ -''' +""" File: leetcode_two_sum.py Created Time: 2022-11-25 Author: Krahets (krahets@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_computational_complexity/space_complexity.py b/codes/python/chapter_computational_complexity/space_complexity.py index 69c1ac015..7efe8d948 100644 --- a/codes/python/chapter_computational_complexity/space_complexity.py +++ b/codes/python/chapter_computational_complexity/space_complexity.py @@ -1,8 +1,8 @@ -''' +""" File: space_complexity.py Created Time: 2022-11-25 Author: Krahets (krahets@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_computational_complexity/time_complexity.py b/codes/python/chapter_computational_complexity/time_complexity.py index a3e698384..bfb5b3b11 100644 --- a/codes/python/chapter_computational_complexity/time_complexity.py +++ b/codes/python/chapter_computational_complexity/time_complexity.py @@ -1,8 +1,8 @@ -''' +""" File: time_complexity.py Created Time: 2022-11-25 Author: Krahets (krahets@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_computational_complexity/worst_best_time_complexity.py b/codes/python/chapter_computational_complexity/worst_best_time_complexity.py index 7b052a3e0..b70dce4f0 100644 --- a/codes/python/chapter_computational_complexity/worst_best_time_complexity.py +++ b/codes/python/chapter_computational_complexity/worst_best_time_complexity.py @@ -1,8 +1,8 @@ -''' +""" File: worst_best_time_complexity.py Created Time: 2022-11-25 Author: Krahets (krahets@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_searching/binary_search.py b/codes/python/chapter_searching/binary_search.py index 3d9ee0ffc..275ae28ec 100644 --- a/codes/python/chapter_searching/binary_search.py +++ b/codes/python/chapter_searching/binary_search.py @@ -1,8 +1,8 @@ -''' +""" File: binary_search.py Created Time: 2022-11-26 Author: timi (xisunyy@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_searching/hashing_search.py b/codes/python/chapter_searching/hashing_search.py index 68cffe64a..f49c703be 100644 --- a/codes/python/chapter_searching/hashing_search.py +++ b/codes/python/chapter_searching/hashing_search.py @@ -1,8 +1,8 @@ -''' +""" File: hashing_search.py Created Time: 2022-11-26 Author: timi (xisunyy@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_searching/linear_search.py b/codes/python/chapter_searching/linear_search.py index 933095653..8ff9e0803 100644 --- a/codes/python/chapter_searching/linear_search.py +++ b/codes/python/chapter_searching/linear_search.py @@ -1,8 +1,8 @@ -''' +""" File: linear_search.py Created Time: 2022-11-26 Author: timi (xisunyy@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_sorting/bubble_sort.py b/codes/python/chapter_sorting/bubble_sort.py index 54260dd96..610e3186c 100644 --- a/codes/python/chapter_sorting/bubble_sort.py +++ b/codes/python/chapter_sorting/bubble_sort.py @@ -1,8 +1,8 @@ -''' +""" File: bubble_sort.py Created Time: 2022-11-25 Author: timi (xisunyy@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_sorting/insertion_sort.py b/codes/python/chapter_sorting/insertion_sort.py index db85515d1..bf8492cdd 100644 --- a/codes/python/chapter_sorting/insertion_sort.py +++ b/codes/python/chapter_sorting/insertion_sort.py @@ -1,8 +1,8 @@ -''' +""" File: insertion_sort.py Created Time: 2022-11-25 Author: timi (xisunyy@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_sorting/merge_sort.py b/codes/python/chapter_sorting/merge_sort.py index 533fb9de7..f3fb538ae 100644 --- a/codes/python/chapter_sorting/merge_sort.py +++ b/codes/python/chapter_sorting/merge_sort.py @@ -1,8 +1,8 @@ -''' +""" File: merge_sort.py Created Time: 2022-11-25 Author: timi (xisunyy@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_sorting/quick_sort.py b/codes/python/chapter_sorting/quick_sort.py index 5ba826bfa..f5ce8120c 100644 --- a/codes/python/chapter_sorting/quick_sort.py +++ b/codes/python/chapter_sorting/quick_sort.py @@ -1,8 +1,8 @@ -''' +""" File: quick_sort.py Created Time: 2022-11-25 Author: timi (xisunyy@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_stack_and_queue/array_queue.py b/codes/python/chapter_stack_and_queue/array_queue.py index 7408f2ba2..947c06091 100644 --- a/codes/python/chapter_stack_and_queue/array_queue.py +++ b/codes/python/chapter_stack_and_queue/array_queue.py @@ -1,8 +1,8 @@ -''' +""" File: array_queue.py Created Time: 2022-12-01 Author: Peng Chen (pengchzn@gmail.com) -''' +""" import os.path as osp import sys diff --git a/codes/python/chapter_stack_and_queue/array_stack.py b/codes/python/chapter_stack_and_queue/array_stack.py index 889a372dc..18aee16b0 100644 --- a/codes/python/chapter_stack_and_queue/array_stack.py +++ b/codes/python/chapter_stack_and_queue/array_stack.py @@ -1,8 +1,8 @@ -''' +""" File: array_stack.py Created Time: 2022-11-29 Author: Peng Chen (pengchzn@gmail.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_stack_and_queue/deque.py b/codes/python/chapter_stack_and_queue/deque.py index 7881a965e..30444c6b3 100644 --- a/codes/python/chapter_stack_and_queue/deque.py +++ b/codes/python/chapter_stack_and_queue/deque.py @@ -1,8 +1,8 @@ -''' +""" File: deque.py Created Time: 2022-11-29 Author: Peng Chen (pengchzn@gmail.com) -''' +""" import os.path as osp import sys diff --git a/codes/python/chapter_stack_and_queue/linkedlist_queue.py b/codes/python/chapter_stack_and_queue/linkedlist_queue.py index 0d4a28d4d..2ab7196d3 100644 --- a/codes/python/chapter_stack_and_queue/linkedlist_queue.py +++ b/codes/python/chapter_stack_and_queue/linkedlist_queue.py @@ -1,8 +1,8 @@ -''' +""" File: linkedlist_queue.py Created Time: 2022-12-01 Author: Peng Chen (pengchzn@gmail.com) -''' +""" import os.path as osp import sys diff --git a/codes/python/chapter_stack_and_queue/linkedlist_stack.py b/codes/python/chapter_stack_and_queue/linkedlist_stack.py index 5ee90353b..21f6ca6af 100644 --- a/codes/python/chapter_stack_and_queue/linkedlist_stack.py +++ b/codes/python/chapter_stack_and_queue/linkedlist_stack.py @@ -1,8 +1,8 @@ -''' +""" File: linkedlist_stack.py Created Time: 2022-11-29 Author: Peng Chen (pengchzn@gmail.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_stack_and_queue/queue.py b/codes/python/chapter_stack_and_queue/queue.py index 740d9b1e1..ce220d7c1 100644 --- a/codes/python/chapter_stack_and_queue/queue.py +++ b/codes/python/chapter_stack_and_queue/queue.py @@ -1,8 +1,8 @@ -''' +""" File: queue.py Created Time: 2022-11-29 Author: Peng Chen (pengchzn@gmail.com) -''' +""" import os.path as osp import sys diff --git a/codes/python/chapter_stack_and_queue/stack.py b/codes/python/chapter_stack_and_queue/stack.py index a8d2a7a50..5e95258fe 100644 --- a/codes/python/chapter_stack_and_queue/stack.py +++ b/codes/python/chapter_stack_and_queue/stack.py @@ -1,8 +1,8 @@ -''' +""" File: stack.py Created Time: 2022-11-29 Author: Peng Chen (pengchzn@gmail.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_tree/binary_search_tree.py b/codes/python/chapter_tree/binary_search_tree.py index f9dd9ec95..0f855f279 100644 --- a/codes/python/chapter_tree/binary_search_tree.py +++ b/codes/python/chapter_tree/binary_search_tree.py @@ -1,8 +1,8 @@ -''' +""" File: binary_search_tree.py Created Time: 2022-11-25 Author: Krahets (krahets@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_tree/binary_tree.py b/codes/python/chapter_tree/binary_tree.py index 78afa868e..81828b5cf 100644 --- a/codes/python/chapter_tree/binary_tree.py +++ b/codes/python/chapter_tree/binary_tree.py @@ -1,8 +1,8 @@ -''' +""" File: binary_tree.py Created Time: 2022-11-25 Author: Krahets (krahets@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_tree/binary_tree_bfs.py b/codes/python/chapter_tree/binary_tree_bfs.py index 43f192206..9ae1b3594 100644 --- a/codes/python/chapter_tree/binary_tree_bfs.py +++ b/codes/python/chapter_tree/binary_tree_bfs.py @@ -1,8 +1,8 @@ -''' +""" File: binary_tree_bfs.py Created Time: 2022-11-25 Author: Krahets (krahets@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/chapter_tree/binary_tree_dfs.py b/codes/python/chapter_tree/binary_tree_dfs.py index f8415ef56..094bd4cb5 100644 --- a/codes/python/chapter_tree/binary_tree_dfs.py +++ b/codes/python/chapter_tree/binary_tree_dfs.py @@ -1,8 +1,8 @@ -''' +""" File: binary_tree_dfs.py Created Time: 2022-11-25 Author: Krahets (krahets@163.com) -''' +""" import sys, os.path as osp sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__)))) diff --git a/codes/python/include/binary_tree.py b/codes/python/include/binary_tree.py index 24acb47d4..de2569e42 100644 --- a/codes/python/include/binary_tree.py +++ b/codes/python/include/binary_tree.py @@ -1,8 +1,8 @@ -''' +""" File: binary_tree.py Created Time: 2021-12-11 Author: Krahets (krahets@163.com) -''' +""" import collections diff --git a/codes/python/include/linked_list.py b/codes/python/include/linked_list.py index f6773c6f2..e2eff9a2c 100644 --- a/codes/python/include/linked_list.py +++ b/codes/python/include/linked_list.py @@ -1,8 +1,8 @@ -''' +""" File: linked_list.py Created Time: 2021-12-11 Author: Krahets (krahets@163.com) -''' +""" class ListNode: """Definition for a singly-linked list node diff --git a/codes/python/include/print_util.py b/codes/python/include/print_util.py index f84d548b4..9f211806c 100644 --- a/codes/python/include/print_util.py +++ b/codes/python/include/print_util.py @@ -1,8 +1,8 @@ -''' +""" File: print_util.py Created Time: 2021-12-11 Author: Krahets (krahets@163.com), msk397 (machangxinq@gmail.com) -''' +""" import copy import queue diff --git a/docs/chapter_computational_complexity/time_complexity.md b/docs/chapter_computational_complexity/time_complexity.md index a482c7ae1..0313c533b 100644 --- a/docs/chapter_computational_complexity/time_complexity.md +++ b/docs/chapter_computational_complexity/time_complexity.md @@ -325,7 +325,7 @@ $T(n)$ 是个一次函数,说明时间增长趋势是线性的,因此易得 $$ T(n) \leq c \cdot f(n) $$ - 则可认为 $f(n)$ 给出了 $T(n)$ 的一个渐近上界,记为 + 则可认为 $f(n)$ 给出了 $T(n)$ 的一个渐近上界,记为 $$ T(n) = O(f(n)) $$ diff --git a/docs/chapter_introduction/what_is_dsa.md b/docs/chapter_introduction/what_is_dsa.md index 12ba333b1..67b3c37e1 100644 --- a/docs/chapter_introduction/what_is_dsa.md +++ b/docs/chapter_introduction/what_is_dsa.md @@ -31,12 +31,23 @@ comments: true - 算法是发挥数据结构优势的舞台。数据结构仅存储数据信息,结合算法才可解决特定问题。 - 算法有对应最优的数据结构。给定算法,一般可基于不同的数据结构实现,而最终执行效率往往相差很大。 -如果将数据结构与算法比作「LEGO 乐高」,数据结构就是乐高「积木」,而算法就是把积木拼成目标形态的一系列「操作步骤」。 - ![relationship_between_data_structure_and_algorithm](what_is_dsa.assets/relationship_between_data_structure_and_algorithm.png)

Fig. 数据结构与算法的关系

+如果将「LEGO 乐高」类比到「数据结构与算法」,那么可以得到下表所示的对应关系。 + +
+ +| 数据结构与算法 | LEGO 乐高 | +| -------------- | ---------------------------------------- | +| 输入数据 | 未拼装的积木 | +| 数据结构 | 积木组织形式,包括形状、大小、连接方式等 | +| 算法 | 把积木拼成目标形态的一系列操作步骤 | +| 输出数据 | 积木模型 | + +
+ !!! tip "约定俗成的简称" 在实际讨论中,我们通常会将「数据结构与算法」直接简称为「算法」。例如,我们熟称的 LeetCode 算法题目,实际上同时考察了数据结构和算法两部分知识。 diff --git a/docs/chapter_tree/binary_tree.assets/array_representation_complete_binary_tree.png b/docs/chapter_tree/binary_tree.assets/array_representation_complete_binary_tree.png new file mode 100644 index 0000000000000000000000000000000000000000..f369d5d580f701597aca4e413332524b8b93a032 GIT binary patch literal 73985 zcmeFZg;!K-`#%f=C?QgUA}!t09g1{Gjx^FWprkY?h$7t`(hLnVbcoX3-QC>`{5I!_ z<2mQM*82y%k1UuiYwvsC_to*a`1)E&8XNN|CK3`7wyex6RU{-d;1bl24gx;JDddoW z4`c^b>6b`Fy`&q!fBa0}$eJlCA~6Bi=tyYD_(&*sw*X&A$WQ-x`0f!oWRm}0t0FV} z`yOD;081p)Kki3D(gr^7{uu(_chCIm6D1A#&lS@^|G66tDGlX6*PwpjZY1wlFFb$` zjCV3x4oFB3f8Bi}M>9Vp0rrg~`|5@I2juN!j1PKX*CGfu+(hS-seH_`EL>P8W#M!1Det)nc)mMS>1zuvPK`7m>O2(qO;WpbG&H%+SvwQlX7;+g3LO7=+kao&1i|2EEue$+ zf{XPvUIw7P(>{8yqv%G0?6EPJ6&Q#)k{g)mzB4gwZCy#BF?c+yU$bM#@aL}ov(7vG zcXINPaDmedd?}Q~@>dpSW@c_~HRUO3j)|Z*w!P+kDTT$w$_fhVlB$iGNX+Tl1?xOK 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zHyYG4Pq`EWrazDfI)vVPR|3bT(%wz5Z}eAtsLa-U^ZPGViOIJyv$F2n%+sRglnYYB zHla*4ZrTEE31~svuLs)akMlYm-gBZuODWLzY}CNnZvVxI2E}0N*D*v&+$L%F=(W#{Q$Opg7TjYO}G-L8@2^ z&g!I>XCs7y&z + +## 二叉树表示方式 * + +我们一般使用二叉树的「链表表示」,即存储单位为结点 `TreeNode` ,结点之间通过指针(引用)相连接。本文前述示例代码展示了二叉树在链表表示下的各项基本操作。 + +那能否可以用「数组表示」二叉树呢?答案是肯定的。先来分析一个简单案例,给定一个「完美二叉树」,将结点按照层序遍历的顺序编号(从 0 开始),那么可以推导得出父结点索引与子结点索引之间的「映射公式」:**设结点的索引为 $i$ ,则该结点的左子结点索引为 $2i + 1$ 、右子结点索引为 $2i + 2$** 。 + +**本质上,映射公式的作用就是链表中的指针**。对于层序遍历序列中的任意结点,我们都可以使用映射公式来访问子结点。因此,可以直接使用层序遍历序列(即数组)来表示完美二叉树。 + +![array_representation_mapping](binary_tree.assets/array_representation_mapping.png) + +然而,完美二叉树只是个例,二叉树中间层往往存在许多空结点(即 `null` ),而层序遍历序列并不包含这些空结点,并且我们无法单凭序列来猜测空结点的数量和分布位置,**即理论上存在许多种二叉树都符合该层序遍历序列**。显然,这种情况无法使用数组来存储二叉树。 + +![array_representation_without_empty](binary_tree.assets/array_representation_without_empty.png) + +为了解决此问题,考虑按照完美二叉树的形式来表示所有二叉树,**即在序列中使用特殊符号来显式地表示“空位”**。如下图所示,这样处理后,序列(数组)就可以唯一表示二叉树了。 + +=== "Java" + + ```java title="" + /* 二叉树的数组表示 */ + // 使用 int 的包装类 Integer ,就可以使用 null 来标记空位 + Integer[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 }; + ``` + +=== "C++" + + ```cpp title="" + /* 二叉树的数组表示 */ + // 为了符合数据类型为 int ,使用 int 最大值标记空位 + // 该方法的使用前提是没有结点的值 = INT_MAX + vector tree = { 1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15 }; + ``` + +=== "Python" + + ```python title="" + “”“ 二叉树的数组表示 ”“” + # 直接使用 None 来表示空位 + tree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15] + ``` + +=== "Go" + + ```go title="" + + ``` + +=== "JavaScript" + + ```js title="" + + ``` + +=== "TypeScript" + + ```typescript title="" + + ``` + +=== "C" + + ```c title="" + + ``` + +=== "C#" + + ```csharp title="" + + ``` + +![array_representation_with_empty](binary_tree.assets/array_representation_with_empty.png) + +回顾「完全二叉树」的满足条件,其只有最底层有空结点,并且最底层的结点尽量靠左,因而所有空结点都一定出现在层序遍历序列的末尾。**因为我们先验地确定了空位的位置,所以在使用数组表示完全二叉树时,可以省略存储“空位”**。“便于使用数组表示”也是完全二叉树受欢迎的原因之一。 + +![array_representation_complete_binary_tree](binary_tree.assets/array_representation_complete_binary_tree.png) + +数组表示有两个优点: 一是不需要存储指针,节省空间;二是可以随机访问结点。然而,当二叉树中的“空位”很多时,数组中只包含很少结点的数据,空间利用率很低。