From a772f21b4264453980063136eb8834cd1b106921 Mon Sep 17 00:00:00 2001 From: lifei <1058235935@qq.com> Date: Thu, 3 Dec 2020 22:48:05 +0800 Subject: [PATCH] =?UTF-8?q?=E5=AE=8C=E6=88=90=E7=BA=BF=E7=B4=A2=E4=BA=8C?= =?UTF-8?q?=E5=8F=89=E6=A0=91?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .vscode/settings.json | 16 + ch2/single-link/README.md | 2 +- ch2/single-link/with.cpp | 2 +- ch2/static-link/README.md | 2 +- ch5/README.md | 29 +- ch5/binary-tree-storage/README.md | 30 ++ ...s-of-sequential-storage-of-binary-tree.png | Bin 0 -> 377164 bytes .../link-storage-of-binary-tree.png | Bin 0 -> 327046 bytes ch5/binary-tree-storage/link.cpp | 11 + .../sequential-storage-of-binary-tree.png | Bin 0 -> 351280 bytes ch5/binary-tree-traversal/README.md | 288 ++++++++++++++++++ .../binary-tree-traversal-example.png | Bin 0 -> 67741 bytes ...lue-binary-tree-with-inorder-traversal.png | Bin 0 -> 115609 bytes ...e-binary-tree-with-postorder-traversal.png | Bin 0 -> 123487 bytes ...ue-binary-tree-with-preorder-traversal.png | Bin 0 -> 125084 bytes .../cueing-of-inorder-clue-binary-tree.png | Bin 0 -> 159182 bytes .../cueing-of-inorder-clue-binary-tree2.png | Bin 0 -> 256910 bytes .../three-kinds-of-clue-binary-trees.png | Bin 0 -> 512239 bytes ch5/binary-tree/README.md | 16 +- 19 files changed, 382 insertions(+), 14 deletions(-) create mode 100644 .vscode/settings.json create mode 100644 ch5/binary-tree-storage/README.md create mode 100644 ch5/binary-tree-storage/disadvantages-of-sequential-storage-of-binary-tree.png create mode 100644 ch5/binary-tree-storage/link-storage-of-binary-tree.png create mode 100644 ch5/binary-tree-storage/link.cpp create mode 100644 ch5/binary-tree-storage/sequential-storage-of-binary-tree.png create mode 100644 ch5/binary-tree-traversal/README.md create mode 100644 ch5/binary-tree-traversal/binary-tree-traversal-example.png create mode 100644 ch5/binary-tree-traversal/clue-binary-tree-with-inorder-traversal.png create mode 100644 ch5/binary-tree-traversal/clue-binary-tree-with-postorder-traversal.png create mode 100644 ch5/binary-tree-traversal/clue-binary-tree-with-preorder-traversal.png create mode 100644 ch5/binary-tree-traversal/cueing-of-inorder-clue-binary-tree.png create mode 100644 ch5/binary-tree-traversal/cueing-of-inorder-clue-binary-tree2.png create mode 100644 ch5/binary-tree-traversal/three-kinds-of-clue-binary-trees.png diff --git a/.vscode/settings.json b/.vscode/settings.json new file mode 100644 index 0000000..f6572ed --- /dev/null +++ b/.vscode/settings.json @@ -0,0 +1,16 @@ +{ + "cSpell.words": [ + "infty", + "inorder", + "lchild", + "ltag", + "nextval", + "nlog", + "postorder", + "preorder", + "rchild", + "rclild", + "rtag", + "struct" + ] +} \ No newline at end of file diff --git a/ch2/single-link/README.md b/ch2/single-link/README.md index be8a2e7..019199f 100644 --- a/ch2/single-link/README.md +++ b/ch2/single-link/README.md @@ -406,7 +406,7 @@ int LocateElem(LinkList L, int e) { p = p->next; } - // 找到后返回该节点指针,否则返回 NULL + // 找到后返回该结点指针,否则返回 NULL return p; } ``` diff --git a/ch2/single-link/with.cpp b/ch2/single-link/with.cpp index 802c6be..9f3dfd9 100644 --- a/ch2/single-link/with.cpp +++ b/ch2/single-link/with.cpp @@ -164,7 +164,7 @@ LNode *LocateElem(LinkList L, int e) { p = p->next; } - // 找到后返回该节点指针,否则返回 NULL + // 找到后返回该结点指针,否则返回 NULL return p; } diff --git a/ch2/static-link/README.md b/ch2/static-link/README.md index cb60e99..02e0dcf 100644 --- a/ch2/static-link/README.md +++ b/ch2/static-link/README.md @@ -91,7 +91,7 @@ for (int i = 1; i < a.length; i++) 1. 找到一个空的结点,存入数据元素。 2. 从头结点出发,找到位序为 $i-1$ 的结点。 -3. 修改新节点的 `next`。 +3. 修改新结点的 `next`。 4. 修改 $i-1$ 结点的 `next`。 ## 5. 删除 diff --git a/ch5/README.md b/ch5/README.md index acdf0ed..ec5de14 100644 --- a/ch5/README.md +++ b/ch5/README.md @@ -2,12 +2,12 @@ ## 1. 树的基本概念 -树是 $n(n \geq 0)$ 个节点的有限集合,$n=0$ 时,称为空树。 +树是 $n(n \geq 0)$ 个结点的有限集合,$n=0$ 时,称为空树。 而任意非空树应满足: 1. 有且仅有一个特定的称为**根**的结点。 -2. 当 $n>0$ 时,其余结点可分为 $m(m \geq 0)$ 个互不相交的有限集合,其中每一个集合本身又是一棵树,称为根节点的**子树**。 +2. 当 $n>0$ 时,其余结点可分为 $m(m \geq 0)$ 个互不相交的有限集合,其中每一个集合本身又是一棵树,称为根结点的**子树**。 $n$ 个结点的树中只有 $n-1$ 条边。 @@ -22,7 +22,7 @@ $n$ 个结点的树中只有 $n-1$ 条边。 - 叶子结点:度为 $0$ 的结点。 - 结点的层次:根结点为第一层(或第零层)。 - 结点的高度:从叶子结点(第一层)开始逐层累加。 -- 结点的深度:从根节点(第一层)开始逐层累加。 +- 结点的深度:从根结点(第一层)开始逐层累加。 - 树的高度(深度)是树中结点的最大层数。 - 有序树和无序树 - 路径:树中两个结点之间的路径是由这两个结点之间所经过的**结点序列**构成的。(树的分支是有向的,即从双亲结点指向孩子结点,所以路径一定是自上而下的。) @@ -60,3 +60,26 @@ $n$ 个结点的树中只有 $n-1$ 条边。 - [平衡二叉树](binary-tree/README.md#34-平衡二叉树) ### 4.4. [二叉树的性质](binary-tree/README.md#4-二叉树的性质) + +### 4.5. [二叉树的存储结构](binary-tree-storage/README.md#二叉树的存储结构) + +- [二叉树的顺序存储](binary-tree-storage/README.md#1-二叉树的顺序存储) +- [二叉树的链式存储](binary-tree-storage/README.md#2-二叉树的链式存储) + +### 4.6. [二叉树的遍历](binary-tree-traversal/README.md#二叉树的遍历) + +- [先序遍历](binary-tree-traversal/README.md#1-先序遍历) +- [中序遍历](binary-tree-traversal/README.md#2-中序遍历) +- [后序遍历](binary-tree-traversal/README.md#3-后序遍历) +- [层次遍历](binary-tree-traversal/README.md#5-层次遍历) +- [由遍历序列构造二叉树](binary-tree-traversal/README.md#6-由遍历序列构造二叉树) + +### 4.7. [线索二叉树](binary-tree-traversal/README.md#7-线索二叉树) + +#### 4.7.1. [线索二叉树的概念](binary-tree-traversal/README.md#71-线索二叉树的概念) + +- 先序线索二叉树 +- 中序线索二叉树 +- 后序线索二叉树 + +#### 4.7.2. [线索二叉树的构造](binary-tree-traversal/README.md#72-线索二叉树的构造) diff --git a/ch5/binary-tree-storage/README.md b/ch5/binary-tree-storage/README.md new file mode 100644 index 0000000..6dd953c --- /dev/null +++ b/ch5/binary-tree-storage/README.md @@ -0,0 +1,30 @@ +# 二叉树的存储结构 + +## 1. 二叉树的顺序存储 + +用一组连续的存储单元依次从上而下、从左至右存储[完全二叉树](../binary-tree/README.md#32-完全二叉树)上的结点元素。 + +![二叉树的顺序存储](sequential-storage-of-binary-tree.png) + +在完全二叉树中依次编号,对于结点 $i$: + +- 若存在左孩子,则编号为 $2i$; +- 若存在右孩子,则编号为 $2i+1$; + +![二叉树的顺序存储的缺点](disadvantages-of-sequential-storage-of-binary-tree.png) + +## 2. 二叉树的链式存储 + +用链表存放一棵二叉树,二叉树的每个结点用链表的一个链结点来存储。 + +```cpp +typedef struct BiTNode +{ + ElemType data; + struct BiTNode *Lchild, *rclild; +} BiTNode, *BiTree; +``` + +含有 $n$ 个结点的二叉链表中,有 $n+1$ 个空链域。 + +![二叉树的链式存储](link-storage-of-binary-tree.png) diff --git a/ch5/binary-tree-storage/disadvantages-of-sequential-storage-of-binary-tree.png b/ch5/binary-tree-storage/disadvantages-of-sequential-storage-of-binary-tree.png new file mode 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z(0{dtrAAlU5E4leuPz9fuygsnljnNHc~uqvF7gX_9c~bUU=xoc&Uwr@9do0Nd_Lcws7sw@4llK(7Um1TIvyKPNR&i^x z)MSo701p*k` zsDbU-bSJT|m$wXyr#%OVkwHcAwuQ$5;VzRu5(0}e3rnEE2%fn|6G1SlInjC$&&=fd z=4BW;=z`*IYz67r-`#EAV2S_ru_rT>rp%TF25Cb(Rb00R2zoB036 zRN?x{+PX2H9iaJ#!W;{^vhlW_LDOODX^cZpd^Z_K>Gc@5zqb65DX5ns=H<#gQ z>h+_T#or5Rbk$0GKX&`7+8GGyP-U3KQj2`cPQ_w0ZHuuhRKB5l?AzrQg*b`&4Vl>q h7-aYIGWT-zl6w5$5A)!Clchild != NULL) + { + EnQueue(Q, p->lchild); + } + if (p->rchild != NULL) + { + EnQueue(Q, p->rchild); + } + } +} +``` + +## 6. 由遍历序列构造二叉树 + +- 先序(或后序)遍历序列+中序遍历序列可以确定一棵二叉树。 +- 而先序遍历序列+后序遍历序列不可以确定一颗二叉树。 + +1. 在先序序列中,第一个结点是根结点; +2. 根结点将中序遍历序列划分为两部分; +3. 然后在先序序列中确定两部分的结点,并且两部分的第一个结点分别为左子树的根结点和右子树的根结点; +4. 在子树中递归重复该过程,边能唯一确定一棵二叉树。 + +> 层次遍历序列+中序遍历序列也可以确定一棵二叉树。 + +## 7. 线索二叉树 + +### 7.1. 线索二叉树的概念 + +线索化: + +1. 若无左子树,则将左指针指向其前驱结点; +2. 若无右子树,则将右指针指向其后继结点。 + +![二叉树遍历举例](binary-tree-traversal-example.png) + +- 先序遍历:$124536$ + +![先序线索二叉树](clue-binary-tree-with-preorder-traversal.png) + +- 中序遍历:$425163$ + +![中序线索二叉树](clue-binary-tree-with-inorder-traversal.png) + +- 后序遍历:$452631$ + +![后序线索二叉树](clue-binary-tree-with-postorder-traversal.png) + +线索二叉树的结点结构 + +| ltag | lchild | data | rclild | rtag | +| ---- | ------ | ---- | ------ | ---- | + +$$ +标志域l(r)tag= +\begin{cases} + 0,l(r)child域指示结点的左(右)孩子\\ + 1,l(r)child域指示结点的前驱(后继) +\end{cases} +$$ + +```cpp +typedef struct ThreadNode +{ + ElemType data; + struct ThreadNode *lchild, *rclild; + int ltag, rtag; +}ThreadNode, *ThreadTree; +``` + +这种结点结构构成的二叉链表作为二叉树的存储结构,称为**线索链表**。 + +![三种线索二叉树](three-kinds-of-clue-binary-trees.png) + +### 7.2. 线索二叉树的构造 + +常用的是**中序线索二叉树**。 + +寻找前驱结点: + +- 若左指针为线索,则其指向结点为前驱结点。 +- 若左指针为左孩子,则其左子树的最右侧结点为前驱结点。 + +寻找后继结点: + +- 若右指针为线索,则其指向结点为后继结点。 +- 若右指针为右孩子,则其右子树的最左侧结点为后继结点。 + +中序线索二叉树线索化 + +```cpp +// p 为线索二叉树的根结点 +// pre 为对应结点的前驱结点 +void InThread(ThreadTree &p, ThreadTree &pre) +{ + if (p != NULL) + { + InThread(p->lchild, pre); + if (p->lchild == NULL) + { + p->lchild = pre; + p->ltag = 1; + } + if (pre != NULL && pre->lchild == NULL) + { + p->rchild = p; + p->rtag = 1; + } + InThread(p->rchild, pre); + } +} +``` + +```cpp +void CreateInThread(ThreadTree T) +{ + ThreadTree pre = NULL; + if (T != NULL) + { + InThread(T, pre); + pre->rchild = NULL; + pre->rtag = 1; + } +} +``` + +![中序线索二叉树的线索化](cueing-of-inorder-clue-binary-tree.png) + +![中序线索二叉树的线索化2](cueing-of-inorder-clue-binary-tree2.png) + +```cpp +// p 为线索二叉树的根结点 +ThreadNode *FirstNode(ThreadNode *p) +{ + while (p->ltag == 0) + { + p = p->lchild; + } + return p; +} +``` + +```cpp +// p 为结点 +ThreadNode *NextNode(ThreadNode *p) +{ + if (p->rtag == 0) + { + return FirstNode(p.rchild); // 如果有孩子结点,则后继结点就是右子树最左侧的结点 + } + else + { + return p.rchild; + } +} +``` + +```cpp +// 遍历 +void InOrder(ThreadNode *T) +{ + for (ThreadNode *p = FirstNode(T); p != NULL; p = NextNode(p)) + { + visit(p); + } +} +``` diff --git a/ch5/binary-tree-traversal/binary-tree-traversal-example.png b/ch5/binary-tree-traversal/binary-tree-traversal-example.png new file mode 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