给定一个二叉树，任务是打印给定二叉树中的所有指数级别。
An Exponential Level is a level whose all nodes of that levels equals to xy, & where x is a minimum possible positive constant & y is a variable positive integer.
例子：
Input:
20
/ \
9 81
/ \ / \
3 10 70 243
/ \
81 909
Output:
20
9 81
Explanation:
There are 2 exponential levels:
20: 201 = 20.
9, 81: 32 = 9, 34 = 81.
Input:
8
/ \
4 81
/ \ / \
5 125 625 5
/ \
81 909
Output:
8
5 125 625 5
方法：解决上述问题的主要思想是使用Level Order Tree Traversal。
- 执行给定二叉树的级别顺序遍历并将每个级别存储在一个向量中。
- 那么，在每一层中，如果每个节点都可以表示为 x y的形式，则对于 y ≥ 0。
- 如果该层节点的任何值不等于xy ，则跳到下一层。
- 打印上述条件为真的所有级别。
下面是上述方法的实现：
C++
// C++ program for printing all
// Exponential levels of binary Tree
#include
using namespace std;
int N = 1e6;
// To store all prime numbers
vector prime;
void SieveOfEratosthenes()
{
// Create a boolean array "prime[0..N]" and initialize
// all entries it as true. A value in prime[i] will
// finally be false if i is Not a prime, else true.
bool check[N + 1];
memset(check, true, sizeof(check));
for (int p = 2; p * p <= N; p++) {
// check if prime[p] is not changed,
// then it is a prime
if (check[p] == true) {
prime.push_back(p);
// Update all multiples of p greater than or
// equal to the square of it
// numbers which are multiple of p and are
// less than p^2 are already been marked.
for (int i = p * p; i <= N; i += p)
check[i] = false;
}
}
}
// A Tree node
struct Node {
int key;
struct Node *left, *right;
};
// 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 check
// whether the given node
// equals to x^y for some y>0
bool is_key(int n, int x)
{
double p;
// Take logx(n) with base x
p = log10(n) / log10(x);
int no = (int)(pow(x, int(p)));
if (n == no)
return true;
return false;
}
// Function to find x
int find_x(int n)
{
if (n == 1)
return 1;
double num, den, p;
// Take log10 of n
num = log10(n);
int x, no;
for (int i = 2; i <= n; i++) {
den = log10(i);
// Log(n) with base i
p = num / den;
// Raising i to the power p
no = (int)(pow(i, int(p)));
if (abs(no - n) < 1e-6) {
x = i;
break;
}
}
return x;
}
// Function to check whether Level
// is Exponential or not
bool isLevelExponential(vector& L)
{
// retrieve the value of x
// for that level
int x = find_x(L[0]);
for (int i = 1; i < L.size(); i++) {
// Checking that element is
// equal x^y for some y
if (!is_key(L[i], x))
return false;
}
return true;
}
// Function to print an Exponential level
void printExponentialLevels(vector& Lev)
{
for (auto x : Lev) {
cout << x << " ";
}
cout << endl;
}
// Utility function to get Exponential
// Level of a given Binary tree
void find_ExponentialLevels(struct Node* node,
struct Node* queue[],
int index, int size)
{
vector Lev;
while (index < size) {
int curr_size = size;
while (index < curr_size) {
struct Node* temp = queue[index];
Lev.push_back(temp->key);
// Push left child in a queue
if (temp->left != NULL)
queue[size++] = temp->left;
// Push right child in a queue
if (temp->right != NULL)
queue[size++] = temp->right;
// Increament index
index++;
}
// check if level is exponential
if (isLevelExponential(Lev)) {
printExponentialLevels(Lev);
}
Lev.clear();
}
}
// Function to find total no of nodes
// In a given binary tree
int findSize(struct Node* node)
{
// Base condition
if (node == NULL)
return 0;
return 1
+ findSize(node->left)
+ findSize(node->right);
}
// Function to find Exponential levels
// In a given binary tree
void printExponentialLevels(struct Node* node)
{
int t_size = findSize(node);
// Create queue
struct Node* queue[t_size];
// Push root node in a queue
queue[0] = node;
find_ExponentialLevels(node, queue, 0, 1);
}
// Driver Code
int main()
{
/* 20
/ \
9 81
/ \ / \
3 9 81 243
/ \
81 909 */
// Create Binary Tree as shown
Node* root = newNode(20);
root->left = newNode(9);
root->right = newNode(81);
root->left->left = newNode(3);
root->left->right = newNode(9);
root->right->left = newNode(81);
root->right->right = newNode(243);
root->right->left->left = newNode(81);
root->right->right->right = newNode(909);
// To save all prime numbers
SieveOfEratosthenes();
// Print Exponential Levels
printExponentialLevels(root);
return 0;
} Python3
# Python3 program for printing
# all Exponential levels of
# binary Tree
import math
# A Tree node
class Node:
def __init__(self, key):
self.key = key
self.left = None
self.right = None
# Utility function to create
# a new node
def newNode(key):
temp = Node(key)
return temp
N = 1000000
# Vector to store all the
# prime numbers
prime = []
# Function to store all the
# prime numbers in an array
def SieveOfEratosthenes():
# Create a boolean array "prime[0..N]"
# and initialize all the entries in it
# as true. A value in prime[i]
# will finally be false if
# i is Not a prime, else true.
check = [True for i in range(N + 1)]
p = 2
while(p * p <= N):
# If prime[p] is not
# changed, then it is
# a prime
if (check[p]):
prime.append(p);
# Update all multiples of p
# greater than or equal to
# the square of it
# numbers which are multiples of p
# and are less than p^2
# are already marked.
for i in range(p * p, N + 1, p):
check[i] = False;
p += 1
# Function To check
# whether the given node
# equals to x^y for some y>0
def is_key(n, x):
# Take logx(n) with base x
p = (math.log10(n) /
math.log10(x));
no = int(math.pow(x, int(p)));
if (n == no):
return True;
return False;
# Function to find x
def find_x(n):
if (n == 1):
return 1;
den = 0
p = 0
# Take log10 of n
num = math.log10(n);
x = 0
no = 0;
for i in range(2, n + 1):
den = math.log10(i);
# Log(n) with base i
p = num / den;
# Raising i to the power p
no = int(math.pow(i, int(p)));
if(abs(no - n) < 0.000001):
x = i;
break;
return x;
# Function to check whether Level
# is Exponential or not
def isLevelExponential(L):
# retrieve the value of x
# for that level
x = find_x(L[0]);
for i in range(1, len(L)):
# Checking that element is
# equal x^y for some y
if (not is_key(L[i], x)):
return False;
return True;
# Function to print an
# Exponential level
def printExponentialLevels(Lev):
for x in Lev:
print(x, end = ' ')
print()
# Utility function to get Exponential
# Level of a given Binary tree
def find_ExponentialLevels(node, queue,
index, size):
Lev = []
while (index < size):
curr_size = size;
while index < curr_size:
temp = queue[index];
Lev.append(temp.key);
# Push left child in a queue
if (temp.left != None):
queue[size] = temp.left;
size += 1
# Push right child in a queue
if (temp.right != None):
queue[size] = temp.right;
size += 1
# Increament index
index += 1;
# check if level is exponential
if (isLevelExponential(Lev)):
printExponentialLevels(Lev);
Lev.clear();
# Function to find total no of nodes
# In a given binary tree
def findSize(node):
# Base condition
if (node == None):
return 0;
return (1 + findSize(node.left) +
findSize(node.right));
# Function to find Exponential levels
# In a given binary tree
def printExponentialLevel(node):
t_size = findSize(node);
# Create queue
queue=[0 for i in range(t_size)]
# Push root node in a queue
queue[0] = node;
find_ExponentialLevels(node, queue,
0, 1);
# Driver code
if __name__ == "__main__":
''' 20
/ \
9 81
/ \ / \
3 9 81 243
/ \
81 909 '''
# Create Binary Tree as shown
root = newNode(20);
root.left = newNode(9);
root.right = newNode(81);
root.left.left = newNode(3);
root.left.right = newNode(9);
root.right.left = newNode(81);
root.right.right = newNode(243);
root.right.left.left = newNode(81);
root.right.right.right = newNode(909);
# To save all prime numbers
SieveOfEratosthenes();
# Print Exponential Levels
printExponentialLevel(root);
# This code is contributed by Rutvik_56输出：
20
9 81
3 9 81 243
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