查询子树 DFS 中的第 M 个节点

给定一棵有 N 个节点和 N-1 条边的树。同样给定一个整数 M 和一个节点，任务是打印给定节点的子树的 DFS 中的第 M 个节点以进行多次查询。

注意：M 不会大于给定节点的子树中的节点数。

Input: M = 3, node = 1 
Output: 4 
In the above example if 1 is given as the node, then the DFS of subtree will be 1 2 4 6 7 5 3, hence if M is 3, then the 3rd node is 4 

Input: M = 4, node = 2 
Output: 7 
If 2 is given as the node, then the DFS of the subtree will be 2 4 6 7 5., hence if M is 4 then the 4th node is 7.

方法：

• 在邻接列表中添加节点之间的边。
• 调用 DFS函数生成完整树的 DFS。
• 使用 under[] 数组存储给定节点下的子树的高度，包括该节点。
• 在 DFS函数中，在每次递归调用时不断增加子树的大小。
• 使用散列标记完成的 DFS 中的节点索引。
• 让树的 DFS 中给定节点的索引为ind ，则第 M 个节点将位于索引ind + M -1处，因为节点的子树的 DFS 将始终是从该节点开始的连续子数组。

下面是上述方法的实现。

C++

// C++ program for Queries
// for DFS of subtree of a node in a tree
#include 
using namespace std;
const int N = 100000;
 
// Adjacency list to store the
// tree nodes connection
vector v[N];
 
// stores the index of node in DFS
unordered_map mp;
 
// stores the index of node in
// original node
vector a;
 
// Function to call DFS and count nodes
// under that subtree
void dfs(int under[], int child, int parent)
{
 
    // stores the DFS of tree
    a.push_back(child);
 
    // height of subtree
    under[child] = 1;
 
    // iterate for children
    for (auto it : v[child]) {
 
        // if not equal to parent
        // so that it does not traverse back
        if (it != parent) {
 
            // call DFS for subtree
            dfs(under, it, child);
 
            // add the height
            under[child] += under[it];
        }
    }
}
 
// Function to return the DFS of subtree of node
int printnodeDFSofSubtree(int node, int under[], int m)
{
    // index of node in the original DFS
    int ind = mp[node];
 
    // height of subtree of node
    return a[ind + m - 1];
}
 
// Function to add edges to a tree
void addEdge(int x, int y)
{
    v[x].push_back(y);
    v[y].push_back(x);
}
 
// Marks the index of node in original DFS
void markIndexDfs()
{
    int size = a.size();
 
    // marks the index
    for (int i = 0; i < size; i++) {
        mp[a[i]] = i;
    }
}
 
// Driver Code
int main()
{
    int n = 7;
 
    // add edges of a tree
    addEdge(1, 2);
    addEdge(1, 3);
    addEdge(2, 4);
    addEdge(2, 5);
    addEdge(4, 6);
    addEdge(4, 7);
 
    // array to store the height of subtree
    // of every node in a tree
    int under[n + 1];
 
    // Call the function DFS to generate the DFS
    dfs(under, 1, 0);
 
    // Function call to mark the index of node
    markIndexDfs();
 
    int m = 3;
 
    // Query 1
    cout << printnodeDFSofSubtree(1, under, m) << endl;
 
    // Query 2
    m = 4;
    cout << printnodeDFSofSubtree(2, under, m);
 
    return 0;
}

Java

// Java program for Queries for
// DFS of subtree of a node in a tree
import java.util.*;
 
class GFG{
     
// Adjacency list to store the
// tree nodes connection
static ArrayList> v;
 
// Stores the index of node in DFS
static HashMap mp;
 
// Stores the index of node in
// original node
static ArrayList a;
 
// Function to call DFS and count nodes
// under that subtree
static void dfs(int under[], int child,
                int parent)
{
 
    // Stores the DFS of tree
    a.add(child);
 
    // Height of subtree
    under[child] = 1;
 
    // iterate for children
    for(int it : v.get(child))
    {
 
        // If not equal to parent
        // so that it does not traverse back
        if (it != parent)
        {
 
            // Call DFS for subtree
            dfs(under, it, child);
 
            // Add the height
            under[child] += under[it];
        }
    }
}
 
// Function to return the DFS of subtree of node
static int printnodeDFSofSubtree(int node,
                                 int under[],
                                 int m)
{
     
    // Index of node in the original DFS
    int ind = mp.get(node);
 
    // Height of subtree of node
    return a.get(ind + m - 1);
}
 
// Function to add edges to a tree
static void addEdge(int x, int y)
{
    v.get(x).add(y);
    v.get(y).add(x);
}
 
// Marks the index of node in original DFS
static void markIndexDfs()
{
    int size = a.size();
 
    // Marks the index
    for(int i = 0; i < size; i++)
    {
        mp.put(a.get(i), i);
    }
}
 
// Driver Code
public static void main(String[] args)
{
    int n = 7;
 
    mp = new HashMap<>();
    v = new ArrayList<>();
    a = new ArrayList<>();
     
    for(int i = 0; i < n + 1; i++)
        v.add(new ArrayList<>());
         
    // Add edges of a tree
    addEdge(1, 2);
    addEdge(1, 3);
    addEdge(2, 4);
    addEdge(2, 5);
    addEdge(4, 6);
    addEdge(4, 7);
 
    // Array to store the height of subtree
    // of every node in a tree
    int under[] = new int[n + 1];
     
    // Call the function DFS to generate the DFS
    dfs(under, 1, 0);
 
    // Function call to mark the index of node
    markIndexDfs();
 
    int m = 3;
 
    // Query 1
    System.out.println(
        printnodeDFSofSubtree(1, under, m));
 
    // Query 2
    m = 4;
    System.out.println(
        printnodeDFSofSubtree(2, under, m));
}
}
 
// This code is contributed by jrishabh99

Python3

# Python3 program for Queries
# for DFS of subtree of a node in a tree
N = 100000
 
# Adjacency list to store the
# tree nodes connection
v = [[]for i in range(N)]
 
# stores the index of node in DFS
mp = {}
 
# stores the index of node in
# original node
a = []
 
# Function to call DFS and count nodes
# under that subtree
def dfs(under, child, parent):
     
    # stores the DFS of tree
    a.append(child)
     
    # height of subtree
    under[child] = 1
     
    # iterate for children
    for it in v[child]:
         
        # if not equal to parent
        # so that it does not traverse back
        if (it != parent):
             
            # call DFS for subtree
            dfs(under, it, child)
             
            # add the height
            under[child] += under[it]
             
# Function to return the DFS of subtree of node
def printnodeDFSofSubtree(node, under, m):
     
    # index of node in the original DFS
    ind = mp[node]
     
    # height of subtree of node
    return a[ind + m - 1]
     
# Function to add edges to a tree
def addEdge(x, y):
    v[x].append(y)
    v[y].append(x)
 
# Marks the index of node in original DFS
def markIndexDfs():
     
    size = len(a)
     
    # marks the index
    for i in range(size):
        mp[a[i]] = i
     
# Driver Code
 
n = 7
 
# add edges of a tree
addEdge(1, 2)
addEdge(1, 3)
addEdge(2, 4)
addEdge(2, 5)
addEdge(4, 6)
addEdge(4, 7)
 
# array to store the height of subtree
# of every node in a tree
under = [0]*(n + 1)
 
# Call the function DFS to generate the DFS
dfs(under, 1, 0)
 
# Function call to mark the index of node
markIndexDfs()
 
m = 3
 
# Query 1
print(printnodeDFSofSubtree(1, under, m))
 
# Query 2
m = 4
print(printnodeDFSofSubtree(2, under, m))
 
# This code is contributed by SHUBHAMSINGH10

C#

// C# program for Queries for DFS
// of subtree of a node in a tree
using System;
using System.Collections.Generic;
 
class GFG{
     
// Adjacency list to store the
// tree nodes connection
static List> v;
 
// Stores the index of node in DFS
static Dictionary mp;
 
// Stores the index of node in
// original node
static List a;
 
// Function to call DFS and count nodes
// under that subtree
static void dfs(int []under, int child,
                int parent)
{
     
    // Stores the DFS of tree
    a.Add(child);
 
    // Height of subtree
    under[child] = 1;
 
    // Iterate for children
    foreach(int it in v[child])
    {
         
        // If not equal to parent so
        // that it does not traverse back
        if (it != parent)
        {
             
            // Call DFS for subtree
            dfs(under, it, child);
 
            // Add the height
            under[child] += under[it];
        }
    }
}
 
// Function to return the DFS of subtree of node
static int printnodeDFSofSubtree(int node,
                                 int []under,
                                 int m)
{
     
    // Index of node in the original DFS
    int ind = mp[node];
 
    // Height of subtree of node
    return a[ind + m - 1];
}
 
// Function to add edges to a tree
static void addEdge(int x, int y)
{
    v[x].Add(y);
    v[y].Add(x);
}
 
// Marks the index of node in original DFS
static void markIndexDfs()
{
    int size = a.Count;
 
    // Marks the index
    for(int i = 0; i < size; i++)
    {
        mp.Add(a[i], i);
    }
}
 
// Driver Code
public static void Main(String[] args)
{
    int n = 7;
 
    mp = new Dictionary();
    v = new List>();
    a = new List();
     
    for(int i = 0; i < n + 1; i++)
        v.Add(new List());
         
    // Add edges of a tree
    addEdge(1, 2);
    addEdge(1, 3);
    addEdge(2, 4);
    addEdge(2, 5);
    addEdge(4, 6);
    addEdge(4, 7);
 
    // Array to store the height of subtree
    // of every node in a tree
    int []under = new int[n + 1];
     
    // Call the function DFS to generate the DFS
    dfs(under, 1, 0);
 
    // Function call to mark the index of node
    markIndexDfs();
 
    int m = 3;
 
    // Query 1
    Console.WriteLine(
        printnodeDFSofSubtree(1, under, m));
 
    // Query 2
    m = 4;
    Console.WriteLine(
        printnodeDFSofSubtree(2, under, m));
}
}
 
// This code is contributed by Amit Katiyar

Javascript

 
// Javascript program for Queries for DFS
// of subtree of a node in a tree
   
// Adjacency list to store the
// tree nodes connection
var v = [];
 
// Stores the index of node in DFS
var mp = new Map();
 
// Stores the index of node in
// original node
var a = [];
 
// Function to call DFS and count nodes
// under that subtree
function dfs(under, child, parent)
{
     
    // Stores the DFS of tree
    a.push(child);
 
    // Height of subtree
    under[child] = 1;
 
    // Iterate for children
    for(var it of v[child])
    {
         
        // If not equal to parent so
        // that it does not traverse back
        if (it != parent)
        {
             
            // Call DFS for subtree
            dfs(under, it, child);
 
            // Push the height
            under[child] += under[it];
        }
    }
}
 
// Function to return the DFS of subtree of node
function printnodeDFSofSubtree(node, under, m)
{
     
    // Index of node in the original DFS
    var ind = mp.get(node);
 
    // Height of subtree of node
    return a[ind + m - 1];
}
 
// Function to add edges to a tree
function addEdge(x, y)
{
    v[x].push(y);
    v[y].push(x);
}
 
// Marks the index of node in original DFS
function markIndexDfs()
{
    var size = a.length;
 
    // Marks the index
    for(var i = 0; i < size; i++)
    {
        mp.set(a[i], i);
    }
}
 
// Driver Code
var n = 7;
mp = new Map();
v = [];
a = [];
 
for(var i = 0; i < n + 1; i++)
    v.push(Array());
     
// Push edges of a tree
addEdge(1, 2);
addEdge(1, 3);
addEdge(2, 4);
addEdge(2, 5);
addEdge(4, 6);
addEdge(4, 7);
 
// Array to store the height of subtree
// of every node in a tree
var under = new Array(n + 1);
 
// Call the function DFS to generate the DFS
dfs(under, 1, 0);
 
// Function call to mark the index of node
markIndexDfs();
var m = 3;
 
// Query 1
document.write(printnodeDFSofSubtree(
    1, under, m) + "<br>");
     
// Query 2
m = 4;
document.write(printnodeDFSofSubtree(
    2, under, m));
 
// This code is contributed by rutvik_56
 

输出：

4
7

时间复杂度： O(1)，用于处理每个查询。
辅助空间： O(N)