二叉树简介及C++实现

二叉树是每个结点最多有两个子树的树结构，即结点的度最大为2。通常子树被称作”左子树”和”右子树”。二叉树是一个连通的无环图。

二叉树是递归定义的，其结点有左右子树之分，逻辑上二叉树有五种基本形态：(1)、空二叉树；(2)、只有一个根结点的二叉树；(3)、只有左子树；(4)、只有右子树；(5)、完全二叉树。

二叉树类型：

(1)、满二叉树：深度(层数)为k，且有2^k-1个结点的二叉树。这种树的特点是每一层上的结点数都是最大结点数。即除了叶结点外每一个结点都有左右子树且叶节点都处在最低层。

(2)、完全二叉树：除最后一层外，其余层都是满的，并且最后一层或者是满的，或者是在右边缺少连续若干节点，即叶子结点都是从左到右依次排布。具有n个节点的完全二叉树的深度为floor(log(2n))+1。深度为k的完全二叉树，至少有2^(k-1)个结点，至多有(2^k)-1个结点。

(3)、平衡二叉树：又被称为AVL树，它是一颗二叉排序树，且具有以下性质：它是一颗空树或它的左右两个子树的高度差的绝对值不超过1，并且左右两个子树都是一颗平衡二叉树。

(4)、斜树：所有的结点都只有左子树(左斜树)，或者只有右子树(右斜树)。

(5)、二叉搜素树(或二叉排序树)：特殊的二叉树，每个结点都不比它左子树的任意元素小，而且不比它的右子树的任意元素大。二叉搜索树的左右子树也都是二叉搜索树。按中序遍历，则会得到按升序排列的有序数据集。

二叉树不是树的一种特殊情形。

遍历二叉树：按一定的规则和顺序走遍二叉树的所有结点，使每一个结点都被访问一次，而且只被访问一次。对一颗二叉树的遍历有四种情况：先序遍历、中序遍历、后序遍历、按层遍历。

(1)、先序遍历：先访问根结点，再先序遍历左子树，最后再先序遍历右子树，即先访问根结点-------左子树------右子树。

(2)、中序遍历：先中序遍历左子树，然后再访问根结点，最后再中序遍历右子树，即先访问左子树------根结点------右子树。

(3)、后序遍历：先后序遍历左子树，然后再后序遍历右子树，最后再访问根结点，即先访问左子树------右子树------根结点。

(4)、按层遍历：从上到下，从左到右依次访问结点。

下面代码是二叉搜索树的实现，主要包括树的创建、插入、删除、查找、遍历、保存、载入。

binary_search_tree.hpp:

#ifndef FBC_CPPBASE_TEST_BINARY_SEARCH_TREE_HPP_
#define FBC_CPPBASE_TEST_BINARY_SEARCH_TREE_HPP_#include <vector>
#include <fstream>
#include <string>namespace binary_search_tree_ {typedef struct info {int id; // suppose id is uniquestd::string name;int age;std::string addr;
} info;typedef struct node {info data;node* left = nullptr;node* right = nullptr;
} node;class BinarySearchTree {
public:BinarySearchTree() = default;~BinarySearchTree() { DeleteTree(tree); }typedef std::tuple<int, int, std::string, int, std::string> row; // flag(-1: no node, 0: have a node), id, name, age, addrint Init(const std::vector<info>& infos); // create binary search treebool Search(int id, info& data) const;int Insert(const info& data);int Delete(int id); // delete a nodeint Traversal(int type) const; // 0: pre-order, 1: in-order, 2: post-order, 3: levelint GetMaxDepth() const; // get tree max depthint GetMinDepth() const; // get tree min depthint GetNodesCount() const; // get tree node countbool IsBinarySearchTree() const; // whether or not is a binary search tree//bool IsBinaryBalanceTree() const; // whether ot not is a binary balance treeint GetMinValue(info& data) const;int GetMaxValue(info& data) const;int SaveTree(const char* name) const; // tree write in txt fileint LoadTree(const char* name);protected:void PreorderTraversal(const node* ptr) const;void InorderTraversal(const node* ptr) const;void PostorderTraversal(const node* ptr) const;void LevelTraversal(const node* ptr) const;void LevelTraversal(const node* ptr, int level) const;void DeleteTree(node* ptr);void Insert(node* ptr, const info& data);const node* Search(const node* ptr, int id) const;void IsBinarySearchTree(const node* ptr, bool is_bst) const;int GetNodesCount(const node* ptr) const;int GetMaxDepth(const node* ptr) const;int GetMinDepth(const node* ptr) const;//bool IsBinaryBalanceTree(const node* ptr) const;node* Delete(node* ptr, int id); // return new rootnode* GetMinValue(node* ptr);void NodeToRow(const node* ptr, std::vector<row>& rows, int pos) const;void RowToNode(node* ptr, const std::vector<row>& rows, int n, int pos);private:node* tree = nullptr;bool flag;
};int test_binary_search_tree();} // namespace binary_search_tree_#endif // FBC_CPPBASE_TEST_BINARY_SEARCH_TREE_HPP_

binary_search_tree.cpp:

#include "binary_search_tree.hpp"
#include <set>
#include <iostream>
#include <limits>
#include <tuple>
#include <string>
#include <sstream>
#include <string.h>
#include <algorithm>namespace binary_search_tree_ {int BinarySearchTree::Init(const std::vector<info>& infos)
{std::vector<int> ids;for (const auto& info : infos) {ids.emplace_back(info.id);}std::set<int> id_set(ids.cbegin(), ids.cend());if (id_set.size() != ids.size()) {fprintf(stderr, "id must be unique\n");return -1;}for (const auto& info : infos) {Insert(info);}return 0;
}bool BinarySearchTree::Search(int id, info& data) const
{const node* root = tree;const node* tmp = Search(root, id);if (tmp) {data = tmp->data;return true;} else {return false;}
}const node* BinarySearchTree::Search(const node* ptr, int id) const
{if (ptr) {if (ptr->data.id == id) {return ptr;} else if (ptr->data.id > id) {return Search(ptr->left, id);} else {return Search(ptr->right, id);}} else {return nullptr;}
}int BinarySearchTree::Insert(const info& data)
{flag = true;if (tree) {Insert(tree, data);} else {tree = new node;tree->data = data;tree->left = nullptr;tree->right = nullptr;}return (int)flag;
}void BinarySearchTree::Insert(node* ptr, const info& data)
{if (ptr->data.id == data.id) {flag = false;return;}if (ptr->data.id < data.id) {if (ptr->right) {Insert(ptr->right, data);} else {ptr->right = new node;ptr->right->data = data;ptr->right->left = nullptr;ptr->right->right = nullptr;}} else {if (ptr->left) {Insert(ptr->left, data);} else {ptr->left = new node;ptr->left->data = data;ptr->left->left = nullptr;ptr->left->right = nullptr;}}
}bool BinarySearchTree::IsBinarySearchTree() const
{bool is_bst = true;const node* root = tree;IsBinarySearchTree(root, is_bst);return is_bst;
}void BinarySearchTree::IsBinarySearchTree(const node* ptr, bool is_bst) const
{static int last_data = std::numeric_limits<int>::min();if (!ptr) return;IsBinarySearchTree(ptr->left, is_bst);if (last_data < ptr->data.id) last_data = ptr->data.id;else {is_bst = false;return;}IsBinarySearchTree(ptr->right, is_bst);
}int BinarySearchTree::Traversal(int type) const
{if (!tree) {fprintf(stderr, "Error: it is an empty tree\n");return -1;}const node* root = tree;if (type == 0)PreorderTraversal(root);else if (type == 1)InorderTraversal(root);else if (type == 2)PostorderTraversal(root);else if (type == 3)LevelTraversal(root);else {fprintf(stderr, "Error: don't suppot traversal type, type only support: 0: pre-order, 1: in-order, 2: post-order\n");return -1;}return 0;
}void BinarySearchTree::PreorderTraversal(const node* ptr) const
{if (ptr) {fprintf(stdout, "Info: id: %d, name: %s, age: %d, addr: %s\n",ptr->data.id, ptr->data.name.c_str(), ptr->data.age, ptr->data.addr.c_str());PreorderTraversal(ptr->left);PreorderTraversal(ptr->right);}
}void BinarySearchTree::InorderTraversal(const node* ptr) const
{if (ptr) {InorderTraversal(ptr->left);fprintf(stdout, "Info: id: %d, name: %s, age: %d, addr: %s\n",ptr->data.id, ptr->data.name.c_str(), ptr->data.age, ptr->data.addr.c_str());InorderTraversal(ptr->right);}
}void BinarySearchTree::PostorderTraversal(const node* ptr) const
{if (ptr) {PostorderTraversal(ptr->left);PostorderTraversal(ptr->right);fprintf(stdout, "Info: id: %d, name: %s, age: %d, addr: %s\n",ptr->data.id, ptr->data.name.c_str(), ptr->data.age, ptr->data.addr.c_str());}
}void BinarySearchTree::LevelTraversal(const node* ptr) const
{int h = GetMaxDepth();for (int i = 1; i <= h; ++i)LevelTraversal(ptr, i);
}void BinarySearchTree::LevelTraversal(const node* ptr, int level) const
{if (!ptr) return;if (level == 1)fprintf(stdout, "Info: id: %d, name: %s, age: %d, addr: %s\n",ptr->data.id, ptr->data.name.c_str(), ptr->data.age, ptr->data.addr.c_str());else if (level > 1) {LevelTraversal(ptr->left, level-1);LevelTraversal(ptr->right, level-1);}
}void BinarySearchTree::DeleteTree(node* ptr)
{if (ptr) { DeleteTree(ptr->left);DeleteTree(ptr->right);delete ptr;}
}int BinarySearchTree::GetNodesCount() const
{   const node* root = tree;return GetNodesCount(root);
}int BinarySearchTree::GetNodesCount(const node* ptr) const
{if (!ptr) return 0;else return GetNodesCount(ptr->left) + 1 + GetNodesCount(ptr->right);
}int BinarySearchTree::GetMaxDepth() const
{const node* root = tree;return GetMaxDepth(root);
}int BinarySearchTree::GetMaxDepth(const node* ptr) const
{if (!ptr) return 0;int left_depth = GetMaxDepth(ptr->left);int right_depth = GetMaxDepth(ptr->right);return std::max(left_depth, right_depth) + 1;
}int BinarySearchTree::GetMinDepth() const
{const node* root = tree;return GetMinDepth(root);
}int BinarySearchTree::GetMinDepth(const node* ptr) const
{if (!ptr) return 0;int left_depth = GetMaxDepth(ptr->left);int right_depth = GetMaxDepth(ptr->right);return std::min(left_depth, right_depth) + 1;
}/*bool BinarySearchTree::IsBinaryBalanceTree() const
{const node* root = tree;return IsBinaryBalanceTree(root);
}bool BinarySearchTree::IsBinaryBalanceTree(const node* ptr) const
{// TODO: code need to modifyif (GetMaxDepth(ptr) - GetMinDepth(ptr) <= 1) return true;else return false;
}*/int BinarySearchTree::GetMinValue(info& data) const
{if (!tree) {fprintf(stderr, "Error： it is a empty tree\n");return -1;}const node* root = tree;while (root->left) root = root->left;data = root->data;return 0;
}int BinarySearchTree::GetMaxValue(info& data) const
{if (!tree) {fprintf(stderr, "Error： it is a empty tree\n");return -1;}const node* root = tree;while (root->right) root = root->right;data = root->data;return 0;
}int BinarySearchTree::Delete(int id)
{if (!tree) {fprintf(stderr, "Error: it is a empty tree\n");return -1;}const node* root = tree;const node* ret = Search(root, id);if (!ret) {fprintf(stdout, "Warning: this id don't exist in the tree: %d", id);return 0;}  tree = Delete(tree, id);   return 0;
}node* BinarySearchTree::GetMinValue(node* ptr)
{node* tmp = ptr;while (tmp->left) tmp = tmp->left;return tmp;
}node* BinarySearchTree::Delete(node* ptr, int id)
{if (!ptr) return ptr;if (id < ptr->data.id)ptr->left = Delete(ptr->left, id);else if (id > ptr->data.id) ptr->right = Delete(ptr->right, id);else {if (!ptr->left) {node* tmp = ptr->right;delete ptr;return tmp;} else if (!ptr->right) {node* tmp = ptr->left;delete ptr;return tmp;}node* tmp = GetMinValue(ptr->right);ptr->data = tmp->data;ptr->right = Delete(ptr->right, tmp->data.id);}return ptr;
}int BinarySearchTree::SaveTree(const char* name) const
{std::ofstream file(name, std::ios::out);if (!file.is_open()) {fprintf(stderr, "Error: open file fail: %s\n", name);return -1;}const node* root = tree;int max_depth = GetMaxDepth(root);int max_nodes = (1 << max_depth) -1;root = tree;int nodes_count = GetNodesCount(root);//fprintf(stdout, "max_depth: %d, max nodes: %d\n", max_depth, max_nodes);file<<nodes_count<<","<<max_depth<<std::endl;std::vector<row> vec(max_nodes, std::make_tuple(-1, -1, " ", -1, " "));root = tree;NodeToRow(root, vec, 0);   for (const auto& v : vec) {file << std::get<0>(v)<<","<<std::get<1>(v)<<","<<std::get<2>(v)<<","<<std::get<3>(v)<<","<<std::get<4>(v)<<std::endl;}file.close();return 0;
}void BinarySearchTree::NodeToRow(const node* ptr, std::vector<row>& rows, int pos) const
{if (!ptr) return;rows[pos] = std::make_tuple(0, ptr->data.id, ptr->data.name, ptr->data.age, ptr->data.addr);if (ptr->left) NodeToRow(ptr->left, rows, 2 * pos + 1);if (ptr->right) NodeToRow(ptr->right, rows, 2 * pos + 2);
}int BinarySearchTree::LoadTree(const char* name)
{std::ifstream file(name, std::ios::in);if (!file.is_open()) {fprintf(stderr, "Error: open file fail: %s\n", name);return -1;}std::string line, cell;std::getline(file, line);std::stringstream line_stream(line);std::vector<int> vec;while (std::getline(line_stream, cell, ',')) {vec.emplace_back(std::stoi(cell));}if (vec.size() != 2) {fprintf(stderr, "Error: parse txt file fail\n");return -1;}fprintf(stdout, "nodes count: %d, max depth: %d\n", vec[0], vec[1]);int max_nodes = (1 << vec[1]) - 1;std::vector<row> rows;while (std::getline(file, line)) {std::stringstream line_stream2(line);std::vector<std::string> strs;while (std::getline(line_stream2, cell, ',')) {strs.emplace_back(cell);}if (strs.size() != 5) {fprintf(stderr, "Error: parse line fail\n");return -1;}row tmp = std::make_tuple(std::stoi(strs[0]), std::stoi(strs[1]), strs[2], std::stoi(strs[3]), strs[4]);rows.emplace_back(tmp);}if (rows.size() != max_nodes || std::get<0>(rows[0]) == -1) {fprintf(stderr, "Error: parse txt file line fail\n");return -1;}node* root = new node;root->data = {std::get<1>(rows[0]), std::get<2>(rows[0]), std::get<3>(rows[0]), std::get<4>(rows[0])};root->left = nullptr;root->right = nullptr;tree = root;RowToNode(root, rows, max_nodes, 0);file.close();return 0;
}void BinarySearchTree::RowToNode(node* ptr, const std::vector<row>& rows, int n, int pos)
{if (!ptr || n == 0) return;int new_pos = 2 * pos + 1;if (new_pos < n && std::get<0>(rows[new_pos]) != -1) {ptr->left = new node;ptr->left->data = {std::get<1>(rows[new_pos]), std::get<2>(rows[new_pos]), std::get<3>(rows[new_pos]), std::get<4>(rows[new_pos])};ptr->left->left = nullptr;ptr->left->right = nullptr;RowToNode(ptr->left, rows, n, new_pos);}new_pos = 2 * pos + 2;if (new_pos < n && std::get<0>(rows[new_pos]) != -1) {ptr->right = new node;ptr->right->data = {std::get<1>(rows[new_pos]), std::get<2>(rows[new_pos]), std::get<3>(rows[new_pos]), std::get<4>(rows[new_pos])};ptr->right->left = nullptr;ptr->right->right = nullptr;RowToNode(ptr->right, rows, n, new_pos);}
}int test_binary_search_tree()
{fprintf(stdout, "create binary search tree:\n");std::vector<info> infos{{1004, "Tom", 8, "Beijing"}, {1005, "Jack", 9, "Tianjin"}, {1003, "Mark", 6, "Hebei"}, {1009, "Lisa", 11, "Beijiing"}, {1007, "Piter", 4, "Hebei"}, {1001, "Viner", 6, "Beijing"}};BinarySearchTree bstree;bstree.Init(infos);fprintf(stdout, "\ninsert operation:\n");std::vector<info> infos2{{1007, "xxx", 11, "yyy"}, {1008, "Lorena", 22, "Hebie"}, {1002, "Eillen", 14, "Shanxi"}};for (const auto& info : infos2) {int flag = bstree.Insert(info);if (flag) fprintf(stdout, "insert success\n");else fprintf(stdout, "Warning: id %d already exists, no need to insert\n", info.id);}fprintf(stdout, "\ntraversal operation:\n");fprintf(stdout, "pre-order traversal:\n");bstree.Traversal(0);fprintf(stdout, "in-order traversal:\n");bstree.Traversal(1);fprintf(stdout, "post-order traversal:\n");bstree.Traversal(2);fprintf(stdout, "level traversal:\n");bstree.Traversal(3);fprintf(stdout, "\nsearch operation:\n");std::vector<int> ids {1009, 2000};for (auto id : ids) {info ret;bool flag = bstree.Search(id, ret);if (flag)fprintf(stdout, "found: info: %d, %s, %d, %s\n", ret.id, ret.name.c_str(), ret.age, ret.addr.c_str());elsefprintf(stdout, "no find: no id info: %d\n", id);}fprintf(stdout, "\nwhether or not is a binary search tree operation:\n");bool flag2 = bstree.IsBinarySearchTree();if (flag2) fprintf(stdout, "it is a binary search tree\n");else fprintf(stdout, "it is not a binary search tree\n");fprintf(stdout, "\ncalculate node count operation:\n");int count = bstree.GetNodesCount();fprintf(stdout, "tree node count: %d\n", count);fprintf(stdout, "\ncalculate tree depth operation:\n");int max_depth = bstree.GetMaxDepth();int min_depth = bstree.GetMinDepth();fprintf(stdout, "tree max depth: %d, min depth: %d\n", max_depth, min_depth);/*fprintf(stdout, "\nwhether or not is a binary balance tree operation:\n");flag2 = bstree.IsBinaryBalanceTree();if (flag2) fprintf(stdout, "it is a binary balance tree\n");else fprintf(stdout, "it is not a binary balance tree\n");*/fprintf(stdout, "\nget min and max value(id):\n");info value;bstree.GetMinValue(value);fprintf(stdout, "tree min value: id: %d\n", value.id);bstree.GetMaxValue(value);fprintf(stdout, "tree max value: id: %d\n", value.id);fprintf(stdout, "\ndelete node operation:\n");bstree.Delete(1005);bstree.Traversal(1);fprintf(stdout, "\nsave tree operation:\n");
#ifdef _MSC_VERchar* name = "E:/GitCode/Messy_Test/testdata/binary_search_tree.model";
#elsechar* name = "testdata/binary_search_tree.model";
#endifbstree.SaveTree(name);fprintf(stdout, "\nload tree operation:\n");BinarySearchTree bstree2;bstree2.LoadTree(name);int count2 = bstree2.GetNodesCount();int max_depth2 = bstree2.GetMaxDepth();int min_depth2 = bstree2.GetMinDepth();fprintf(stdout, "tree node count: %d, tree max depth: %d, min depth: %d\n", count2, max_depth2, min_depth2);bstree2.Traversal(1);return 0;
}} // namespace binary_search_tree_

支持Linux和Windows直接编译，Windows通过VS，linux下执行prj/linux_cmake_CppBaseTest/build.sh脚本。执行结果如下：

保存的binary_search_tree.model结果如下：

7,4
0,1004,Tom,8,Beijing
0,1003,Mark,6,Hebei
0,1009,Lisa,11,Beijiing
0,1001,Viner,6,Beijing
-1,-1, ,-1,
0,1007,Piter,4,Hebei
-1,-1, ,-1,
-1,-1, ,-1,
0,1002,Eillen,14,Shanxi
-1,-1, ,-1,
-1,-1, ,-1,
-1,-1, ,-1,
0,1008,Lorena,22,Hebie
-1,-1, ,-1,
-1,-1, ,-1,

GitHub： https://github.com/fengbingchun/Messy_Test

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