二叉树中序遍历，先序遍历，后序遍历（递归栈，非递归栈，Morris Traversal）

例题

• 中序遍历94. Binary Tree Inorder Traversal
• 先序遍历144. Binary Tree Preorder Traversal
• 后序遍历145. Binary Tree Postorder Traversal

递归栈

递归函数栈的方法很基础，写法也很简单，三种遍历方式之间只需要改变一行代码的位置即可

中序遍历

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     TreeNode *left;
 *     TreeNode *right;
 *     TreeNode(int x) : val(x), left(NULL), right(NULL) {}
 * };
 */
class Solution {
public:
    
    void inorder(TreeNode* root, vector<int>& v){
        if(root != nullptr) {
            inorder(root->left, v);
            v.push_back(root->val); // 改变位置的代码
            inorder(root->right, v);
        }
    }
    vector<int> inorderTraversal(TreeNode* root) {
        vector<int> v;
        
        inorder(root, v);
        return v;
    }
};

先序遍历

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     TreeNode *left;
 *     TreeNode *right;
 *     TreeNode(int x) : val(x), left(NULL), right(NULL) {}
 * };
 */
class Solution {
public:
    
    void inorder(TreeNode* root, vector<int>& v){
        if(root != nullptr) {
            v.push_back(root->val);
            inorder(root->left, v);
            inorder(root->right, v);
        }
    }
    vector<int> inorderTraversal(TreeNode* root) {
        vector<int> v;
        
        inorder(root, v);
        return v;
    }
};

后序遍历

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     TreeNode *left;
 *     TreeNode *right;
 *     TreeNode(int x) : val(x), left(NULL), right(NULL) {}
 * };
 */
class Solution {
public:
    
    void inorder(TreeNode* root, vector<int>& v){
        if(root != nullptr) {
            inorder(root->left, v);
            inorder(root->right, v);
            v.push_back(root->val);
        }
    }
    vector<int> inorderTraversal(TreeNode* root) {
        vector<int> v;
        
        inorder(root, v);
        return v;
    }
};

非递归栈

当树的深度过大时，函数栈可能会溢出，这时候需要我们使用数据结构中的栈，来模拟节点在栈中的压入和弹出

中序遍历

cur指针时刻指向需要处理的节点
如果当前节点不为空，则压入栈中，并改变cur指针指向其左节点
如果当前节点为空，也代表空节点的中序遍历自动完成，无需压栈，这时候需要弹出栈顶的节点，保存栈顶节点的值，并更改cur指向其右子树，以完成右子树的中序遍历

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     TreeNode *left;
 *     TreeNode *right;
 *     TreeNode(int x) : val(x), left(NULL), right(NULL) {}
 * };
 */
class Solution {
public:
    vector<int> inorderTraversal(TreeNode* root) {
        vector<int> v;
        stack<TreeNode*> s;
        TreeNode* cur = root;

        while(cur || !s.empty()){
            if(cur){
                s.push(cur);
                cur = cur->left;
            } else {
                cur = s.top();
                s.pop();
                v.push_back(cur->val);
                cur = cur->right;
            }
        }
        
        return v;
    }
};

先序遍历

先序遍历与中序遍历代码相比只改变了保存节点的值的代码的位置，当先访问根的时候就记录节点，而不是等左子树遍历完，弹出根节点的时候再记录

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     TreeNode *left;
 *     TreeNode *right;
 *     TreeNode(int x) : val(x), left(NULL), right(NULL) {}
 * };
 */
class Solution {
public:
    vector<int> preorderTraversal(TreeNode* root) {
        vector<int> v;
        stack<TreeNode*> s;
        TreeNode* cur = root;

        while(cur || !s.empty()){
            if(cur){
                v.push_back(cur->val);
                s.push(cur);
                cur = cur->left;
            } else {
                cur = s.top();
                s.pop();
                cur = cur->right;
            }
        }
        
        return v;
    }
};

后序遍历

因为后序遍历的压栈顺序是左-右-根，由于先遍历完左子树，然后遍历完右子树，然后才能处理当前节点，为了和之前的代码的结构保持一致，我们可以反向处理，也就是按根-右-左的顺序压栈，
结果反向输出即可

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     TreeNode *left;
 *     TreeNode *right;
 *     TreeNode(int x) : val(x), left(NULL), right(NULL) {}
 * };
 */
class Solution {
public:
    vector<int> postorderTraversal(TreeNode* root) {
        vector<int> v;
        stack<TreeNode*> s;
        TreeNode* cur = root;

        while(cur || !s.empty()){
            if(cur){
                v.push_back(cur->val);
                s.push(cur);
                cur = cur->right;
            } else {
                cur = s.top();
                s.pop();
                cur = cur->left;
            }
        }
        
        reverse(v.begin(), v.end()); // 反向输出结果
        return v;
    }
};

Morris Traversal

线索二叉树，O(n)时间，常数空间
Morris Traversal方法遍历二叉树（非递归，不用栈，O(1)空间）
就是当前节点的中序遍历前驱节点，如果此前驱节点的右指针为空，则将此前驱节点的右指针指向当前节点
当寻找当前节点的中序遍历前驱节点时，发现能循环到自己，说明左子树已经遍历完，需要遍历右子树

中序遍历

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     TreeNode *left;
 *     TreeNode *right;
 *     TreeNode(int x) : val(x), left(NULL), right(NULL) {}
 * };
 */
class Solution {
public:
    vector<int> inorderTraversal(TreeNode* root) {
        TreeNode* cur, *prev;
        vector<int> v;
        cur = root;
        prev = nullptr;
        
        while(cur){
            if(cur->left == nullptr){
                v.push_back(cur->val);
                cur = cur->right;
            } else {
                prev = cur->left;
                while(prev->right != nullptr && prev->right != cur) 
                    prev = prev->right;
                
                if(prev->right == nullptr){
                    prev->right = cur;
                    cur = cur->left;
                } else {
                    v.push_back(cur->val);
                    prev->right = nullptr;
                    cur = cur->right;
                }
            }
        }
        
        return v;
    }
};

先序遍历

同样先序遍历与中序遍历相比也只需要改变一行代码的位置

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     TreeNode *left;
 *     TreeNode *right;
 *     TreeNode(int x) : val(x), left(NULL), right(NULL) {}
 * };
 */
class Solution {
public:
    vector<int> preorderTraversal(TreeNode* root) {
        TreeNode* cur, *prev;
        vector<int> v;
        cur = root;
        prev = nullptr;
        
        while(cur){
            if(cur->left == nullptr){
                v.push_back(cur->val);
                cur = cur->right;
            } else {
                prev = cur->left;
                while(prev->right != nullptr && prev->right != cur) 
                    prev = prev->right;
                
                if(prev->right == nullptr){
                    v.push_back(cur->val);
                    prev->right = cur;
                    cur = cur->left;
                } else {
                    prev->right = nullptr;
                    cur = cur->right;
                }
            }
        }
        
        return v;
    }
};

后序遍历

后序遍历需要反向输出cur->left到cur的中序前驱结点之间的路径

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     TreeNode *left;
 *     TreeNode *right;
 *     TreeNode(int x) : val(x), left(NULL), right(NULL) {}
 * };
 */
class Solution {
public:

    void reverse_traverse_reverse(TreeNode* cur, vector<int>& v){
        TreeNode* prev = nullptr;

        while(cur){
            auto right = cur->right;
            cur->right = prev;
            prev = cur;
            cur = right;
        }


        cur = prev;
        prev = nullptr;

        while(cur){
            auto right = cur->right;
            cur->right = prev;
            v.push_back(cur->val);
            prev = cur;
            cur = right;
        }
    }
    vector<int> postorderTraversal(TreeNode* root) {
        TreeNode* cur, *prev;
        vector<int> v;
        cur = new TreeNode(-1);
        prev = nullptr;
        cur->left = root;
        while(cur){
            if(cur->left == nullptr){
                cur = cur->right;
            } else {
                prev = cur->left;
                while(prev->right != nullptr && prev->right != cur)
                    prev = prev->right;
                
                if(prev->right == nullptr){
                    prev->right = cur;
                    cur = cur->left;
                } else {
                    prev->right = nullptr;
                    reverse_traverse_reverse(cur->left, v);
                    cur = cur->right;
                }
            }
        }
        
        return v;
    }
};

总结

三种遍历方式中，后序遍历的处理比较麻烦，但是无论是使用递归栈，非递归栈还是Morris Traversal，代码的结构都是一样的，中序和前序甚至只有一行代码位置的差别
使用栈的版本中，最坏空间复杂度为O（n）链型，平均空间复杂度为O（lgn）
Morris Traversal空间复杂度为O(1)