Middle To Up-Down Order traversal of a Binary Tree

Last Updated : 10 May, 2021

Given a binary tree, the task is to traverse this binary tree from the middle to the up-down order.
In Middle to up-down order traversal, the following steps are performed:

First, print the middle level of the tree.
Then, print the elements at one level above the middle level of the tree.
Then, print the elements at one level below the middle level of the tree.
Then, print the elements at two levels above the middle level of the tree.
Then, print the elements at two levels below the middle level of the tree and so on.

Note: In this problem, the middle level is considered at ((H / 2) + 1)^th level where H is the height of the tree and the level of the root is considered as 1.
Examples:

Input:
                 10 
               /     \ 
              12     13 
              /       \ 
             14        15 
             / \       / \ 
            21 22     23 24  
Output:
        14, 15, 
        12, 13, 
        21, 22, 23, 24, 
        10,
Explanation:
There are 4 levels in the tree. 
Therefore, Middle level = ((4 / 2) + 1) = 3
Now, the tree is traversed in the following way:
Middle level: 14, 15
One level above the middle level: 12, 13
One level below the middle level: 21, 22, 23, 24
Two levels above the middle level: 10

Input:
                  5 
                /   \ 
              2      13 
            /   \      \
           4    25       6
          /      / \ 
        11       3 21 
                     \
                      9
Output:
        4, 25, 6 
        2, 13 
        11, 3, 21
        5
        9
Explanation:
There are 5 levels in the tree. 
Therefore, Middle level = ((5 / 2) + 1) = 3.
Now, the tree is traversed in the following way:
Middle level: 4, 25, 6
One level above the middle level: 2, 13
One level below the middle level: 11, 3, 21
Two levels above the middle level: 5
Two levels below the middle level: 9

Approach: The idea is to store the tree in a matrix. In order to do that,

The height and width of the tree are computed.
After computing the height and maximum width, a 2d matrix is created and a breadth-first search is performed on the tree to store the tree in the matrix.
For example:

Then, simply iterate over the matrix and print the values from the matrix as per the given condition.

Below is the implementation of the above approach:

C++

// C++ program to traverse the tree
// from the middle to up and down
 
#include <bits/stdc++.h>
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 calculate the
// height of the tree
int findHeight(struct Node* node)
{
    // Base condition
    if (node == NULL)
        return 0;
    int leftHeight = findHeight(node->left);
    int rightHeight = findHeight(node->right);
 
    // Return the maximum of the height
    // of the left and right subtree
    return 1 + (leftHeight > rightHeight
                    ? leftHeight
                    : rightHeight);
}
 
// Function to find the width of the tree
void findWidth(struct Node* node, int& maxValue,
               int& minValue, int hd)
{
    // Base cases
    if (node == NULL)
        return;
 
    if (hd > maxValue) {
        maxValue = hd;
    }
 
    if (hd < minValue) {
        minValue = hd;
    }
 
    // Recursively call the function twice to find
    // the maximum width
    findWidth(node->left, maxValue, minValue, hd - 1);
    findWidth(node->right, maxValue, minValue, hd + 1);
}
 
// Function to traverse the tree and
// store level order traversal in a matrix
void BFS(int** mtrx, struct Node* node)
{
    // Create queue for storing
    // the addresses of nodes
    queue<struct Node*> qu;
 
    qu.push(node);
 
    int i = -1, j = -1;
 
    struct Node* poped_node = NULL;
 
    // Iterating over the queue to perform
    // breadth-first search traversal
    while (!qu.empty()) {
 
        i++;
 
        int qsize = qu.size();
 
        while (qsize--) {
            j++;
 
            poped_node = qu.front();
 
            // Store the data of the node into
            // the matrix
            mtrx[i][j] = poped_node->key;
            qu.pop();
 
            // Performing BFS for the remaining
            // nodes in the queue
            if (poped_node->left != NULL) {
                qu.push(poped_node->left);
            }
 
            if (poped_node->right != NULL) {
                qu.push(poped_node->right);
            }
        }
 
        j = -1;
    }
}
 
// Function for Middle to Up Down
// Traversal of Binary Tree
void traverse_matrix(int** mtrx, int height,
                     int width)
{
    // Variables to handle rows and columns
    // of the matrix
    int up = (height / 2);
    int down = up + 1;
 
    bool flag = true;
    int k = 0;
 
    // Print the middle row
    for (int j = 0; j < width; j++) {
        if (mtrx[up][j] != INT_MAX) {
            cout << mtrx[up][j] << ", ";
        }
    }
    cout << endl;
    up--;
 
    // Loop to print the remaining rows
    for (int i = 0; i < (height - 1); i++) {
 
        // Condition to manage up and
        // down indices in the matrix
        if (flag) {
            k = up;
            up--;
            flag = !flag;
        }
 
        else {
            k = down;
            down++;
            flag = !flag;
        }
 
        // Loop to print the value
        // of matrix cells in perticular row
        for (int j = 0; j < width; j++) {
 
            if (mtrx[k][j] != INT_MAX) {
                cout << mtrx[k][j] << ", ";
            }
        }
        cout << endl;
    }
}
 
// A utility function for middle to
// up down traversal
void printPattern(struct Node* node)
{
    // max, min has been taken for
    // calculating the width of tree
    int max_value = INT_MIN;
    int min_value = INT_MAX;
    int hd = 0;
 
    // Calculate the width of a tree
    findWidth(node, max_value, min_value, hd);
    int width = max_value + abs(min_value);
 
    // Calculate the height of the tree
    int height = findHeight(node);
 
    // Double pointer to create 2D array
    int** mtrx = new int*[height];
 
    // Initialize the width for
    // each row of the matrix
    for (int i = 0; i < height; i++) {
        mtrx[i] = new int[width];
    }
 
    // Initialize complete matrix with
    // MAX INTEGER(purpose garbage)
    for (int i = 0; i < height; i++) {
        for (int j = 0; j < width; j++) {
            mtrx[i][j] = INT_MAX;
        }
    }
 
    // Store the BFS traversal of the tree
    // into the 2-D matrix
    BFS(mtrx, node);
 
    // Print the circular clockwise spiral
    // traversal of the tree
    traverse_matrix(mtrx, height, width);
 
    // release extra memory
    // allocated for matrix
    free(mtrx);
}
 
// Driver Code
int main()
{
    /*
                  10
               /     \
              12     13
              /       \
             14        15
             / \       / \
            21 22     23 24 
    */
 
    // Creating the above tree
    Node* root = newNode(10);
    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);
 
    printPattern(root);
 
    return 0;
}

Java

// Java program to traverse the tree
// from the middle to up and down
import java.util.*;
 
class GFG{
 
// A Tree node
static class Node {
    int key;
    Node left, right;
};
static int maxValue, minValue;
 
// Utility function to create
// a new node
static Node newNode(int key)
{
    Node temp = new Node();
    temp.key = key;
    temp.left = temp.right = null;
    return (temp);
}
 
// Function to calculate the
// height of the tree
static int findHeight(Node node)
{
    // Base condition
    if (node == null)
        return 0;
    int leftHeight = findHeight(node.left);
    int rightHeight = findHeight(node.right);
 
    // Return the maximum of the height
    // of the left and right subtree
    return 1 + (leftHeight > rightHeight
                    ? leftHeight
                    : rightHeight);
}
 
// Function to find the width of the tree
static void findWidth(Node node, int hd)
{
    // Base cases
    if (node == null)
        return;
 
    if (hd > maxValue) {
        maxValue = hd;
    }
 
    if (hd < minValue) {
        minValue = hd;
    }
 
    // Recursively call the function twice to find
    // the maximum width
    findWidth(node.left, hd - 1);
    findWidth(node.right, hd + 1);
}
 
// Function to traverse the tree and
// store level order traversal in a matrix
static void BFS(int [][]mtrx, Node node)
{
    // Create queue for storing
    // the addresses of nodes
    Queue<Node> qu = new LinkedList<Node>();
 
    qu.add(node);
 
    int i = -1, j = -1;
 
    Node poped_node = null;
 
    // Iterating over the queue to perform
    // breadth-first search traversal
    while (!qu.isEmpty()) {
 
        i++;
 
        int qsize = qu.size();
 
        while (qsize-- > 0) {
            j++;
 
            poped_node = qu.peek();
 
            // Store the data of the node into
            // the matrix
            mtrx[i][j] = poped_node.key;
            qu.remove();
 
            // Performing BFS for the remaining
            // nodes in the queue
            if (poped_node.left != null) {
                qu.add(poped_node.left);
            }
 
            if (poped_node.right != null) {
                qu.add(poped_node.right);
            }
        }
 
        j = -1;
    }
}
 
// Function for Middle to Up Down
// Traversal of Binary Tree
static void traverse_matrix(int [][]mtrx, int height,
                    int width)
{
    // Variables to handle rows and columns
    // of the matrix
    int up = (height / 2);
    int down = up + 1;
 
    boolean flag = true;
    int k = 0;
 
    // Print the middle row
    for (int j = 0; j < width; j++) {
        if (mtrx[up][j] != Integer.MAX_VALUE) {
            System.out.print(mtrx[up][j]+ ", ");
        }
    }
    System.out.println();
    up--;
 
    // Loop to print the remaining rows
    for (int i = 0; i < (height - 1); i++) {
 
        // Condition to manage up and
        // down indices in the matrix
        if (flag) {
            k = up;
            up--;
            flag = !flag;
        }
 
        else {
            k = down;
            down++;
            flag = !flag;
        }
 
        // Loop to print the value
        // of matrix cells in perticular row
        for (int j = 0; j < width; j++) {
 
            if (mtrx[k][j] != Integer.MAX_VALUE) {
                System.out.print(mtrx[k][j]+ ", ");
            }
        }
        System.out.println();
    }
}
 
// A utility function for middle to
// up down traversal
static void printPattern(Node node)
{
    // max, min has been taken for
    // calculating the width of tree
    maxValue = Integer.MIN_VALUE;
    minValue = Integer.MAX_VALUE;
    int hd = 0;
 
    // Calculate the width of a tree
    findWidth(node, hd);
    int width = maxValue + Math.abs(minValue);
 
    // Calculate the height of the tree
    int height = findHeight(node);
     
    // Double pointer to create 2D array
    int [][]mtrx = new int[width][height];
 
    // Initialize complete matrix with
    // MAX INTEGER(purpose garbage)
    for (int i = 0; i < height; i++) {
        for (int j = 0; j < width; j++) {
            mtrx[i][j] = Integer.MAX_VALUE;
        }
    }
 
    // Store the BFS traversal of the tree
    // into the 2-D matrix
    BFS(mtrx, node);
 
    // Print the circular clockwise spiral
    // traversal of the tree
    traverse_matrix(mtrx, height, width);
 
    // release extra memory
    // allocated for matrix
    mtrx =null;
}
 
// Driver Code
public static void main(String[] args)
{
    /*
                10
            / \
            12 13
            / \
            14 15
            / \ / \
            21 22 23 24
    */
 
    // Creating the above tree
    Node root = newNode(10);
    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);
 
    printPattern(root);
}
}
 
// This code is contributed by Rajput-Ji

Python3

# Python3 program for the
# above approach
from collections import deque
 
# A Tree node
class Node:
   
    def __init__(self, x):
       
        self.key = x
        self.left = None
        self.right = None
 
minValue, maxValue = 0, 0
 
# Function to calculate the
# height of the tree
def findHeight(node):
   
    # Base condition
    if (node == None):
        return 0
    leftHeight = findHeight(node.left)
    rightHeight = findHeight(node.right)
 
    # Return the maximum of the height
    # of the left and right subtree
    if leftHeight > rightHeight:
        return leftHeight + 1
 
    return rightHeight + 1
 
# Function to find the width
# of the tree
def findWidth(node, hd):
    
    global minValue, maxValue
     
    # Base cases
    if (node == None):
        return
 
    if (hd > maxValue):
        maxValue = hd
 
    if (hd < minValue):
        minValue = hd
 
    # Recursively call the
    # function twice to find
    # the maximum width
    findWidth(node.left,
              hd - 1)
    findWidth(node.right,
              hd + 1)
 
# Function to traverse the
# tree and store level order
# traversal in a matrix
def BFS(mtrx, node):
   
    # Create queue for storing
    # the addresses of nodes
    qu = deque()
 
    qu.append(node)
    i = -1
    j = -1
    poped_node = None
 
    # Iterating over the queue
    # to perform breadth-first
    # search traversal
    while (len(qu) > 0):
        i += 1
        qsize = len(qu)
        while (qsize):
            j += 1
            poped_node = qu.popleft()
 
            # Store the data of the
            # node into the matrix
            mtrx[i][j] = poped_node.key
            #qu.pop()
 
            # Performing BFS for
            # the remaining nodes
            # in the queue
            if (poped_node.left != None):
                qu.append(poped_node.left)
 
            if (poped_node.right != None):
                qu.append(poped_node.right)
            qsize -= 1
 
        j = -1
 
#Function for Middle to Up Down
#Traversal of Binary Tree
def traverse_matrix(mtrx,
                    height,width):
   
    # Variables to handle rows
    # and columns of the matrix
    up = (height // 2)
    down = up + 1
 
    flag = True
    k = 0
 
    # Print middle row
    for j in range(width):
        if (mtrx[up][j] != 10 ** 9):
            print(mtrx[up][j], end = ", ")
    print()
    up -= 1
 
    # Loop to print the remaining rows
    for i in range(height - 1):
 
        # Condition to manage up and
        # down indices in the matrix
        if (flag):
            k = up
            up -= 1
            flag = not flag
        else:
            k = down
            down += 1
            flag = not flag
 
        # Loop to print the value
        # of matrix cells in
        # particular row
        for j in range(width):
            if (mtrx[k][j] != 10 ** 9):
                print(mtrx[k][j], end = ", ")
        print()
 
# A utility function for middle to
# up down traversal
def printPattern(node):
   
    global maxValue, minValue
     
    # max, min has been taken for
    # calculating the width of tree
    maxValue = -10 ** 9
    minValue = 10 ** 9
    hd = 0
 
    # Calculate the width
    # of a tree
    findWidth(node, hd)
    width = maxValue + abs(minValue)
 
    # Calculate the height of the tree
    height = findHeight(node)
     
    # print(height,width)
 
    #Double pointer to create 2D array
    mtrx = [[10 ** 9 for i in range(width + 1)]
                     for i in range(height + 1)]
 
    # Store the BFS traversal of the tree
    # into the 2-D matrix
    BFS(mtrx, node)
 
    # Print circular clockwise spiral
    # traversal of the tree
    traverse_matrix(mtrx, height, width)
 
# Driver Code
if __name__ == '__main__':
   
    # /*
    #               10
    #            /     \
    #           12     13
    #           /       \
    #          14        15
    #          / \       / \
    #         21 22     23 24
    # */
 
    # Creating the above tree
    root = Node(10)
    root.left = Node(12)
    root.right = Node(13)
 
    root.right.left = Node(14)
    root.right.right = Node(15)
 
    root.right.left.left = Node(21)
    root.right.left.right = Node(22)
    root.right.right.left = Node(23)
    root.right.right.right = Node(24)
 
    printPattern(root)
 
# This code is contributed by Mohit Kumar 29

C#

// C# program to traverse the tree
// from the middle to up and down
using System;
using System.Collections;
using System.Collections.Generic;
 
class GFG
{
  
// A Tree node
class Node
{
    public int key;
    public Node left, right;
};
 
static int maxValue, minValue;
  
// Utility function to create
// a new node
static Node newNode(int key)
{
    Node temp = new Node();
    temp.key = key;
    temp.left = temp.right = null;
    return (temp);
}
  
// Function to calculate the
// height of the tree
static int findHeight(Node node)
{
    // Base condition
    if (node == null)
        return 0;
    int leftHeight = findHeight(node.left);
    int rightHeight = findHeight(node.right);
  
    // Return the maximum of the height
    // of the left and right subtree
    return 1 + (leftHeight > rightHeight
                    ? leftHeight
                    : rightHeight);
}
  
// Function to find the width of the tree
static void findWidth(Node node, int hd)
{
    // Base cases
    if (node == null)
        return;
  
    if (hd > maxValue) {
        maxValue = hd;
    }
  
    if (hd < minValue) {
        minValue = hd;
    }
  
    // Recursively call the function twice to find
    // the maximum width
    findWidth(node.left, hd - 1);
    findWidth(node.right, hd + 1);
}
  
// Function to traverse the tree and
// store level order traversal in a matrix
static void BFS(int [,]mtrx, Node node)
{
    // Create queue for storing
    // the addresses of nodes
    Queue qu = new Queue(); 
    qu.Enqueue(node);
    int i = -1, j = -1;
    Node poped_node = null;
  
    // Iterating over the queue to perform
    // breadth-first search traversal
    while (qu.Count != 0)
    {
        i++;
        int qsize = qu.Count;
        while (qsize-- > 0) {
            j++;
  
            poped_node = (Node)qu.Peek();
  
            // Store the data of the node into
            // the matrix
            mtrx[i,j] = poped_node.key;
            qu.Dequeue();
  
            // Performing BFS for the remaining
            // nodes in the queue
            if (poped_node.left != null) {
                qu.Enqueue(poped_node.left);
            }
  
            if (poped_node.right != null) {
                qu.Enqueue(poped_node.right);
            }
        }
  
        j = -1;
    }
}
  
// Function for Middle to Up Down
// Traversal of Binary Tree
static void traverse_matrix(int [,]mtrx, int height,
                    int width)
{
    // Variables to handle rows and columns
    // of the matrix
    int up = (height / 2);
    int down = up + 1;
  
    bool flag = true;
    int k = 0;
  
    // Print the middle row
    for (int j = 0; j < width; j++) {
        if (mtrx[up, j] != 100000000) {
            Console.Write(mtrx[up,j]+ ", ");
        }
    }
    Console.WriteLine();
    up--;
  
    // Loop to print the remaining rows
    for (int i = 0; i < (height - 1); i++)
    {
  
        // Condition to manage up and
        // down indices in the matrix
        if (flag) {
            k = up;
            up--;
            flag = !flag;
        }
  
        else {
            k = down;
            down++;
            flag = !flag;
        }
  
        // Loop to print the value
        // of matrix cells in perticular row
        for (int j = 0; j < width; j++)
        {
  
            if (mtrx[k, j] != 100000000) {
                Console.Write(mtrx[k,j]+ ", ");
            }
        }
        Console.WriteLine();
    }
}
  
// A utility function for middle to
// up down traversal
static void printPattern(Node node)
{
    // max, min has been taken for
    // calculating the width of tree
    maxValue = -100000000;
    minValue = 100000000;
    int hd = 0;
  
    // Calculate the width of a tree
    findWidth(node, hd);
    int width = maxValue + Math.Abs(minValue);
  
    // Calculate the height of the tree
    int height = findHeight(node);
      
    // Double pointer to create 2D array
    int [,]mtrx = new int[width,height];
  
    // Initialize complete matrix with
    // MAX INTEGER(purpose garbage)
    for (int i = 0; i < height; i++) {
        for (int j = 0; j < width; j++) {
            mtrx[i, j] = 100000000;
        }
    }
  
    // Store the BFS traversal of the tree
    // into the 2-D matrix
    BFS(mtrx, node);
  
    // Print the circular clockwise spiral
    // traversal of the tree
    traverse_matrix(mtrx, height, width);
  
    // release extra memory
    // allocated for matrix
    mtrx = null;
}
  
// Driver Code
public static void Main(string[] args)
{
    /*
                10
            / \
            12 13
            / \
            14 15
            / \ / \
            21 22 23 24
    */
  
    // Creating the above tree
    Node root = newNode(10);
    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);
  
    printPattern(root);
}
}
 
// This code is contributed by rutvik_56

Output:

14, 15, 
12, 13, 
21, 22, 23, 24, 
10,

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