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 unique
	std::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, addr

	int Init(const std::vector<info>& infos); // create binary search tree
	bool Search(int id, info& data) const;
	int Insert(const info& data);
	int Delete(int id); // delete a node
	int Traversal(int type) const; // 0: pre-order, 1: in-order, 2: post-order, 3: level
	int GetMaxDepth() const; // get tree max depth
	int GetMinDepth() const; // get tree min depth
	int GetNodesCount() const; // get tree node count
	bool IsBinarySearchTree() const; // whether or not is a binary search tree
	//bool IsBinaryBalanceTree() const; // whether ot not is a binary balance tree
	int GetMinValue(info& data) const;
	int GetMaxValue(info& data) const;
	int SaveTree(const char* name) const; // tree write in txt file
	int 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 root
	node* 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 modify
	if (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());
		else
			fprintf(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_VER
	char* name = "E:/GitCode/Messy_Test/testdata/binary_search_tree.model";
#else
	char* name = "testdata/binary_search_tree.model";
#endif
	bstree.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,