编写一个程序来找到一棵树的最大深度或高度

给定一棵二叉树，求它的高度。空树的高度为-1，有一个节点的树的高度为0，下面的树的高度为2。

递归计算节点的左右子树的高度，并将高度分配给节点，作为两个孩子的高度的最大值加1。有关详细信息，请参见下面的伪代码和程序。
算法：

maxDepth()
1. If tree is empty then return -1
2. Else
     (a) Get the max depth of left subtree recursively  i.e., 
          call maxDepth( tree->left-subtree)
     (a) Get the max depth of right subtree recursively  i.e., 
          call maxDepth( tree->right-subtree)
     (c) Get the max of max depths of left and right 
          subtrees and add 1 to it for the current node.
         max_depth = max(max dept of left subtree,  
                             max depth of right subtree) 
                             + 1
     (d) Return max_depth

有关上述示例树的递归函数maxDepth() 的执行，请参见下图。

maxDepth('1') = max(maxDepth('2'), maxDepth('3')) + 1
                               = 1 + 1
                                  /    \
                                /         \
                              /             \
                            /                 \
                          /                     \
               maxDepth('2') = 1                maxDepth('3') = 0
= max(maxDepth('4'), maxDepth('5')) + 1
= 1 + 0   = 1         
                   /    \
                 /        \
               /            \
             /                \
           /                    \
 maxDepth('4') = 0     maxDepth('5') = 0

执行：

C++

// C++ program to find height of tree
#include 
using namespace std;
 
 
/* A binary tree node has data, pointer to left child
and a pointer to right child */
class node
{
    public:
    int data;
    node* left;
    node* right;
};
 
/* Compute the "maxDepth" of a tree -- the number of
    nodes along the longest path from the root node
    down to the farthest leaf node.*/
int maxDepth(node* node)
{
    if (node == NULL)
        return -1;
    else
    {
        /* compute the depth of each subtree */
        int lDepth = maxDepth(node->left);
        int rDepth = maxDepth(node->right);
     
        /* use the larger one */
        if (lDepth > rDepth)
            return(lDepth + 1);
        else return(rDepth + 1);
    }
}
 
/* Helper function that allocates a new node with the
given data and NULL left and right pointers. */
node* newNode(int data)
{
    node* Node = new node();
    Node->data = data;
    Node->left = NULL;
    Node->right = NULL;
     
    return(Node);
}
     
// Driver code   
int main()
{
    node *root = newNode(1);
 
    root->left = newNode(2);
    root->right = newNode(3);
    root->left->left = newNode(4);
    root->left->right = newNode(5);
     
    cout << "Height of tree is " << maxDepth(root);
    return 0;
}
 
// This code is contributed by Amit Srivastav

C

#include 
#include 
 
/* A binary tree node has data, pointer to left child
   and a pointer to right child */
struct node {
    int data;
    struct node* left;
    struct node* right;
};
 
/* Compute the "maxDepth" of a tree -- the number of
    nodes along the longest path from the root node
    down to the farthest leaf node.*/
int maxDepth(struct node* node)
{
    if (node == NULL)
        return -1;
    else {
        /* compute the depth of each subtree */
        int lDepth = maxDepth(node->left);
        int rDepth = maxDepth(node->right);
 
        /* use the larger one */
        if (lDepth > rDepth)
            return (lDepth + 1);
        else
            return (rDepth + 1);
    }
}
 
/* Helper function that allocates a new node with the
   given data and NULL left and right pointers. */
struct node* newNode(int data)
{
    struct node* node
        = (struct node*)malloc(sizeof(struct node));
    node->data = data;
    node->left = NULL;
    node->right = NULL;
 
    return (node);
}
 
int main()
{
    struct node* root = newNode(1);
 
    root->left = newNode(2);
    root->right = newNode(3);
    root->left->left = newNode(4);
    root->left->right = newNode(5);
 
    printf("Height of tree is %d", maxDepth(root));
 
    getchar();
    return 0;
}

Java

// Java program to find height of tree
  
// A binary tree node
class Node
{
    int data;
    Node left, right;
  
    Node(int item)
    {
        data = item;
        left = right = null;
    }
}
  
class BinaryTree
{
     Node root;
  
    /* Compute the "maxDepth" of a tree -- the number of
       nodes along the longest path from the root node
       down to the farthest leaf node.*/
    int maxDepth(Node node)
    {
        if (node == null)
            return -1;
        else
        {
            /* compute the depth of each subtree */
            int lDepth = maxDepth(node.left);
            int rDepth = maxDepth(node.right);
  
            /* use the larger one */
            if (lDepth > rDepth)
                return (lDepth + 1);
             else
                return (rDepth + 1);
        }
    }
      
    /* Driver program to test above functions */
    public static void main(String[] args)
    {
        BinaryTree tree = new BinaryTree();
  
        tree.root = new Node(1);
        tree.root.left = new Node(2);
        tree.root.right = new Node(3);
        tree.root.left.left = new Node(4);
        tree.root.left.right = new Node(5);
  
        System.out.println("Height of tree is : " +
                                      tree.maxDepth(tree.root));
    }
}
 
// This code has been contributed by Amit Srivastav

Python3

# Python3 program to find the maximum depth of tree
 
# A binary tree node
class Node:
 
    # Constructor to create a new node
    def __init__(self, data):
        self.data = data
        self.left = None
        self.right = None
 
# Compute the "maxDepth" of a tree -- the number of nodes
# along the longest path from the root node down to the
# farthest leaf node
def maxDepth(node):
    if node is None:
        return -1 ;
 
    else :
 
        # Compute the depth of each subtree
        lDepth = maxDepth(node.left)
        rDepth = maxDepth(node.right)
 
        # Use the larger one
        if (lDepth > rDepth):
            return lDepth+1
        else:
            return rDepth+1
 
 
# Driver program to test above function
root = Node(1)
root.left = Node(2)
root.right = Node(3)
root.left.left = Node(4)
root.left.right = Node(5)
 
 
print ("Height of tree is %d" %(maxDepth(root)))
 
# This code is contributed by Amit Srivastav

C#

// C# program to find height of tree
using System;
 
// A binary tree node
public class Node
{
    public int data;
    public Node left, right;
 
    public Node(int item)
    {
        data = item;
        left = right = null;
    }
}
 
public class BinaryTree
{
    Node root;
 
    /* Compute the "maxDepth" of a tree -- the number of
    nodes along the longest path from the root node
    down to the farthest leaf node.*/
    int maxDepth(Node node)
    {
        if (node == null)
            return -1;
        else
        {
            /* compute the depth of each subtree */
            int lDepth = maxDepth(node.left);
            int rDepth = maxDepth(node.right);
 
            /* use the larger one */
            if (lDepth > rDepth)
                return (lDepth + 1);
            else
                return (rDepth + 1);
        }
    }
     
    /* Driver code */
    public static void Main(String[] args)
    {
        BinaryTree tree = new BinaryTree();
 
        tree.root = new Node(1);
        tree.root.left = new Node(2);
        tree.root.right = new Node(3);
        tree.root.left.left = new Node(4);
        tree.root.left.right = new Node(5);
 
        Console.WriteLine("Height of tree is : " +
                                    tree.maxDepth(tree.root));
    }
}
 
// This code has been contributed by
// Correction done by Amit Srivastav

Javascript

 
// JavaScript program to find height of tree
 
// A binary tree node
class Node
{
    constructor(item)
    {
        this.data=item;
        this.left=this.right=null;
    }
}
 
    let root;
     
     /* Compute the "maxDepth" of a tree -- the number of
       nodes along the longest path from the root node
       down to the farthest leaf node.*/
    function maxDepth(node)
    {
        if (node == null)
            return -1;
        else
        {
            /* compute the depth of each subtree */
            let lDepth = maxDepth(node.left);
            let rDepth = maxDepth(node.right);
   
            /* use the larger one */
            if (lDepth > rDepth)
                return (lDepth + 1);
             else
                return (rDepth + 1);
        }
    }
     
    /* Driver program to test above functions */
     
        root = new Node(1);
        root.left = new Node(2);
        root.right = new Node(3);
        root.left.left = new Node(4);
        root.left.right = new Node(5);
   
        document.write("Height of tree is : " +
                                      maxDepth(root));
 
 
 
 
// This code is contributed by rag2127
//Correction done by Amit Srivastav
 

---

输出

Height of tree is 2

时间复杂度： O(n)（请参阅我们的帖子树遍历了解详细信息）


方法2：解决这个问题的另一种方法是做Level Order Traversal。在进行级别顺序遍历时，在将每个级别的节点添加到队列中时，我们必须添加NULL 节点，以便每当遇到它时，我们可以增加变量的值并计算该级别。

执行：

C++

#include 
#include 
using namespace std;
 
// A Tree node
struct Node
{
    int key;
    struct Node* left, *right;
};
   
// Utility function to create a new node
Node* newNode(int key)
{
    Node* temp = new Node;
    temp->key = key;
    temp->left = temp->right = NULL;
    return (temp);
}
   
/*Function to find the height(depth) of the tree*/
int height(struct Node* root){
 
    //Initialising a variable to count the
      //height of tree
      int depth = 0;
   
    queueq;
     
      //Pushing first level element along with NULL
      q.push(root);
    q.push(NULL);
    while(!q.empty()){
        Node* temp = q.front();
        q.pop();
       
          //When NULL encountered, increment the value
        if(temp == NULL){
            depth++;
        }
           
          //If NULL not encountered, keep moving
        if(temp != NULL){
            if(temp->left){
                  q.push(temp->left);
            }
              if(temp->right){
                q.push(temp->right);
            }
        }
       
          //If queue still have elements left,
          //push NULL again to the queue.
        else if(!q.empty()){
            q.push(NULL);
        }
    }
    return depth;
}
 
// Driver program
int main()
{
    // Let us create Binary Tree shown in above example
    Node *root  = newNode(1);
    root->left  = newNode(12);
    root->right = newNode(13);
   
    root->right->left   = newNode(14);
    root->right->right  = newNode(15);
   
    root->right->left->left   = newNode(21);
    root->right->left->right  = newNode(22);
    root->right->right->left  = newNode(23);
    root->right->right->right = newNode(24);
   
      cout<<"Height(Depth) of tree is: "<

时间复杂度： O(n)

空间复杂度： O(n)

参考：
http://cslibrary.stanford.edu/110/BinaryTrees.html