二叉树的基本操作

常用的二叉树的链式存储结构有二叉链表和三叉链表来表示，其数据结构的C语言定义以及示意图如下：

本来介绍基于二叉链表的存储结构上的二叉树的几个常用的操作：

  1.二叉树的创建，

  2.使用递归算法进行二叉树的先序，中序和后序遍历。

  3.使用非递归算法进行二叉树的中序遍历（u需要借助于栈）

  4.借助于数据结构队列实现二叉树的层序遍历。

  5.一些其他的函数，求取叶子节点的数目等等。

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1二叉树的创建

使用对树的先序遍历进行输入，叶子节点的左右孩子（其实为空）使用#来代替：

例如对于下图的二叉树，在创建时，其正确的输入应该是+A##/*B##C##D##

Tree createBinaryTree(){ElementType c;Tree new;scanf("%c", &c);if( '#' == c )return NULL;else{if( !(new = (Tree)malloc(sizeof(struct TreeNode))) ){printf("malloc error");exit(0);}new->value = c;new->left = createBinaryTree();new->right = createBinaryTree();}}</span>

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2 递归算法对二叉树进行遍历(先序，中序，后序)

  1.先序

void inOrder(Tree tree){if( NULL == tree )return ;else{inOrder(tree->left);printf("%c ", tree->value);inOrder(tree->right);}}</span>

  2 中序

void preOrder(Tree tree){if( NULL == tree )return ;else{printf("%c ",tree->value);preOrder(tree->left);preOrder(tree->right);}}</span>

  3 后序

void postOrder(Tree tree){if( NULL == tree )return ;else{postOrder(tree->left);postOrder(tree->right);printf("%c ", tree->value);}}</span>

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3.使用非递归算法进行二叉树的中序遍历（需要借助于栈）

   首先给出栈的实现，包括栈的定义，建栈，出栈，入栈。

1栈的定义

struct Stack;typedef struct Stack* PtrToStack;struct Stack{int capacity;int topPosition;Tree* array;};</span>

2相关的栈的操作

PtrToStack createStack(){PtrToStack ptrToStack;if(!(ptrToStack = (PtrToStack)malloc(sizeof(struct Stack)))){printf("malloc error");exit(0);}ptrToStack->capacity = INIT_STACK_SIZE;ptrToStack->topPosition = 0;if( !(ptrToStack->array = (PtrToTreeNode*)malloc(sizeof(PtrToTreeNode) * INIT_STACK_SIZE)) ){printf("malloc error");exit(0);}return ptrToStack;}int push(PtrToStack ptrToStack, Tree tree){if( ptrToStack->capacity - 1 == ptrToStack->topPosition )return 0;ptrToStack->array[ptrToStack->topPosition++] = tree;return 1;}Tree pop(PtrToStack ptrToStack){if( ptrToStack->topPosition == 0 )return 0;ptrToStack->topPosition--;return ptrToStack->array[ptrToStack->topPosition];}</span>

3非递归的中序遍历

void iteration_inOrder(Tree tree){PtrToTreeNode node = tree;PtrToStack stack;stack = createStack();for(;;){for( ;node;node = node->left){push(stack, node);}node = pop(stack);if(!node)break;printf("%c ",node->value);node = node->right;}}</span>

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4.借助于数据结构队列实现二叉树的层序遍历。

1 队列的定义

struct QueueNode;struct QueueLink;typedef struct QueueNode* PtrQueueNode;typedef struct QueueLink* PtrQueueLink;struct QueueNode{Tree treeNode;PtrQueueNode next;};struct QueueLink{PtrQueueNode front;PtrQueueNode rear;};</span>

2 队列的基本操作

PtrQueueLink createQueue(){PtrQueueLink ptrQueueLink;if( !(ptrQueueLink = (PtrQueueLink)malloc(sizeof(struct QueueLink))) ){printf("malloc error");exit(0);}if( !(ptrQueueLink->front = (PtrQueueNode)malloc(sizeof(struct QueueNode))) ){printf("malloc error");exit(0);}ptrQueueLink->rear = ptrQueueLink->front;ptrQueueLink->front->treeNode = NULL;ptrQueueLink->front->next = NULL;return ptrQueueLink;}void enterQueue(PtrQueueLink queue, Tree tree){PtrQueueNode new;if( !(new = (PtrQueueNode)malloc(sizeof(struct QueueNode))) ){printf("malloc error");exit(0);}new->treeNode = tree;new->next = queue->rear->next;queue->rear->next = new;queue->rear = new;}Tree deleteQueue(PtrQueueLink queue){if( queue->front == queue->rear )return NULL;PtrQueueNode del = queue->front->next;queue->front->next = del->next;if( del == queue->rear )queue->rear = queue->front;return del->treeNode;}</span>

3 层序遍历

<span style="font-family:Courier New;font-size:14px;">void levelTraverse(Tree tree){PtrQueueLink queue;queue = createQueue();Tree node = tree;enterQueue(queue, node);while( queue->front != queue->rear ){node = deleteQueue(queue);printf("%c ", node->value);if( node->left )enterQueue(queue, node->left);if( node->right )enterQueue(queue, node->right);}}</span>

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5.一些其他的函数，求取叶子节点的数目等等。

1 求叶子节点数

int leafs(Tree tree){if( !tree )return 0;else if( tree->left == NULL && tree->right == NULL )return 1;elsereturn leafs(tree->left) + leafs(tree->right);}</span>

2 求数的高度

int depth(Tree tree){int left, right;if( !tree )return 0;else{left = depth(tree->left);right = depth(tree->right);return left > right ? left + 1 : right + 1; }}</span>

源码

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#include<stdio.h>#include<stdlib.h>#define INIT_STACK_SIZE 10struct TreeNode;typedef struct TreeNode* PtrToTreeNode;typedef PtrToTreeNode Tree;typedef char ElementType;struct TreeNode{ElementType value;Tree left,right;};struct Stack;typedef struct Stack* PtrToStack;struct Stack{int capacity;int topPosition;Tree* array;};struct QueueNode;struct QueueLink;typedef struct QueueNode* PtrQueueNode;typedef struct QueueLink* PtrQueueLink;struct QueueNode{Tree treeNode;PtrQueueNode next;};struct QueueLink{PtrQueueNode front;PtrQueueNode rear;};Tree createBinaryTree();void inOrder(Tree tree);void preOrder(Tree tree);void postOrder(Tree tree);PtrToStack createStack();int push(PtrToStack ptrToStack, Tree tree);Tree pop(PtrToStack ptrToStack);void iteration_inOrder(Tree tree);int leafs(Tree tree);int depth(Tree tree);PtrQueueLink createQueue();void enterQueue(PtrQueueLink queue, Tree tree);Tree deleteQueue(PtrQueueLink queue);void levelTraverse(Tree tree);//+A##/*B##C##D##int main(int argc, char** argv){Tree tree;printf("please enter binary tree(by preorder):\n");tree = createBinaryTree();printf("recursive inorder:");inOrder(tree);printf("\nrecursive preorder:");preOrder(tree);printf("\nrecursive postorder:");postOrder(tree);printf("\niteration inorder:");iteration_inOrder(tree);printf("\nthe number of leafs is %d\n", leafs(tree));printf("the depth of tree is %d\n", depth(tree));printf("level:");levelTraverse(tree);return 0;}/* * 创建二叉树,要求先序输入二叉树 * */Tree createBinaryTree(){ElementType c;Tree new;scanf("%c", &c);if( '#' == c )return NULL;else{if( !(new = (Tree)malloc(sizeof(struct TreeNode))) ){printf("malloc error");exit(0);}new->value = c;new->left = createBinaryTree();new->right = createBinaryTree();}}/* * 线序遍历二叉树（递归） * */void inOrder(Tree tree){if( NULL == tree )return ;else{inOrder(tree->left);printf("%c ", tree->value);inOrder(tree->right);}}void preOrder(Tree tree){if( NULL == tree )return ;else{printf("%c ",tree->value);preOrder(tree->left);preOrder(tree->right);}}void postOrder(Tree tree){if( NULL == tree )return ;else{postOrder(tree->left);postOrder(tree->right);printf("%c ", tree->value);}}void iteration_inOrder(Tree tree){PtrToTreeNode node = tree;PtrToStack stack;stack = createStack();for(;;){for( ;node;node = node->left){push(stack, node);}node = pop(stack);if(!node)break;printf("%c ",node->value);node = node->right;}}PtrToStack createStack(){PtrToStack ptrToStack;if(!(ptrToStack = (PtrToStack)malloc(sizeof(struct Stack)))){printf("malloc error");exit(0);}ptrToStack->capacity = INIT_STACK_SIZE;ptrToStack->topPosition = 0;if( !(ptrToStack->array = (PtrToTreeNode*)malloc(sizeof(PtrToTreeNode) * INIT_STACK_SIZE)) ){printf("malloc error");exit(0);}return ptrToStack;}int push(PtrToStack ptrToStack, Tree tree){if( ptrToStack->capacity - 1 == ptrToStack->topPosition )return 0;ptrToStack->array[ptrToStack->topPosition++] = tree;return 1;}Tree pop(PtrToStack ptrToStack){if( ptrToStack->topPosition == 0 )return 0;ptrToStack->topPosition--;return ptrToStack->array[ptrToStack->topPosition];}int leafs(Tree tree){if( !tree )return 0;else if( tree->left == NULL && tree->right == NULL )return 1;elsereturn leafs(tree->left) + leafs(tree->right);}int depth(Tree tree){int left, right;if( !tree )return 0;else{left = depth(tree->left);right = depth(tree->right);return left > right ? left + 1 : right + 1; }}PtrQueueLink createQueue(){PtrQueueLink ptrQueueLink;if( !(ptrQueueLink = (PtrQueueLink)malloc(sizeof(struct QueueLink))) ){printf("malloc error");exit(0);}if( !(ptrQueueLink->front = (PtrQueueNode)malloc(sizeof(struct QueueNode))) ){printf("malloc error");exit(0);}ptrQueueLink->rear = ptrQueueLink->front;ptrQueueLink->front->treeNode = NULL;ptrQueueLink->front->next = NULL;return ptrQueueLink;}void enterQueue(PtrQueueLink queue, Tree tree){PtrQueueNode new;if( !(new = (PtrQueueNode)malloc(sizeof(struct QueueNode))) ){printf("malloc error");exit(0);}new->treeNode = tree;new->next = queue->rear->next;queue->rear->next = new;queue->rear = new;}Tree deleteQueue(PtrQueueLink queue){if( queue->front == queue->rear )return NULL;PtrQueueNode del = queue->front->next;queue->front->next = del->next;if( del == queue->rear )queue->rear = queue->front;return del->treeNode;}/* * 层次遍历 * */void levelTraverse(Tree tree){PtrQueueLink queue;queue = createQueue();Tree node = tree;enterQueue(queue, node);while( queue->front != queue->rear ){node = deleteQueue(queue);printf("%c ", node->value);if( node->left )enterQueue(queue, node->left);if( node->right )enterQueue(queue, node->right);}}</span>