Binary tree level order traversal in recursive

 Goal or Objective: Traverse all nodes in a level-wise manner (top to bottom, left to right).

Normally done using a queue (BFS), but here we'll explore the recursive version, its importance, and the math theory behind it.

✅ What Is Level Order Traversal?

Given this binary tree:

151      /   \     152     153    / \   /   154   155  156

 

Level Order Traversal Output: 151 → 152  153 → 154  155  6

🔁 Java Implementation of Recursive Level Order Traversal

Step 1: Find or Evaluate the Height (Depth) of the Tree

java

public class TreeNode {     int information;     TreeNode leftSide, rightSide;     TreeNode(int information) {         this.information= information;         this.leftSide= null;         this.rightSide= null;     } }  public class KcmLevelOrder {     // Step 1: Driver method     void kcmLevelOrder(TreeNode k) {         int h = kcmHeight(k);         for (int i = 1; i <= h; i++) {             kcmPrintLevel(k, i);         }     }     // Step 2: Compute height of tree     int kcmHeight(TreeNode k) {         if (k == null) {             return 0;         } else {             int leftHeight = kcmHeight(k.leftSide);             int rightHeight = kcmHeight(k.rightSide);             return java.lang.Math.max(leftHeight, rightHeight) + 1;         }     }     // Step 3: Print all nodes at a given level     void kcmPrintLevel(TreeNode k, int level) {         if (k == null) return;         if (level == 1) {             System.out.print(" "+k.information+ " ");         } else if (level > 1) {             kcmPrintLevel(k.leftSide, level - 1);             kcmPrintLevel(k.rightSide, level - 1);         }     }

  

Why Recursive Level Order Traversal Is Important

✅ 1. Teaches Tree Depth Understanding

• It forces understanding of how tree height works.
• Every level is a "distance" from the root.

✅ 2. Foundation for BFS Problems

• BFS often uses iteration + queue, but recursion helps:
• Conceptual clarity
• Functional thinking

✅ 3. No Queue Needed

• Queue-based methods are iterative.
• Recursive level-order is a pure divide-and-conquer approach.

Mathematical Theorem Behind It

🔷 1. Tree Height = Max Depth

• Recursive level-order depends on height h of the tree:
• A tree with height h will have h levels.
• Each level i is traversed recursively.

🔷 2. Recursion Uses Mathematical Induction

Level Order Traversal recursively uses inductive logic:

Base Case: If tree is null, return.
Recursive Case: Print all nodes at current level, then move deeper.

This maps directly to mathematical induction:

• P(1): Process level 1
• P(k): If you can process level k, you can process level k+1

🔷 3. Time Complexity Analysis:

• For height h and n nodes:
• printLevel(root, i) takes O(n) time over all levels
• So, overall time: O(n²) in worst case (e.g., skewed trees)
• Better for balanced trees

Comparison Table: Recursive vs Iterative Level Order

Feature	Recursive	Iterative (Queue-based BFS)
Memory	O(h) stack space	O(w) queue (w = max width)
Speed	Slower on unbalanced trees	Faster O(n)
Readability	Very clean and elegant	More control but verbose
Math Basis	Induction, depth recursion	FIFO logic, graph traversal

 

 

package com.kartik.org; public class BinaryTreeLevelOrder {  public static class TreeElementNode  {   int data;   TreeElementNode kleft;   TreeElementNode kright;   TreeElementNode(int data)   {    this.data=data;   }  }  // prints data in level order from kleft to kright   public static boolean kcmLevelOrderTraversalLeftToRight(TreeElementNode kStartNode,int klevel) {    if(null == kStartNode){     return Boolean.FALSE;    }    if(klevel == 1){     System.out.println(kStartNode.data+" ");     return Boolean.TRUE;    }    boolean kleft =kcmLevelOrderTraversalLeftToRight(kStartNode.kleft, klevel-1);    boolean kright =kcmLevelOrderTraversalLeftToRight(kStartNode.kright, klevel-1);    return kleft || kright;   }   public static void kcmLevelOrderTraversalLeftToRight(TreeElementNode kStartNode) {    if(null == kStartNode){     return;    }    int level=1;    while(kcmLevelOrderTraversalLeftToRight(kStartNode, level++)){    }   }   //print data kright to kleft   public static boolean kcmLevelOrderTraversalRightToLeft(TreeElementNode kStartNode,int klevel) {    if(null == kStartNode){     return Boolean.FALSE;    }    if(klevel == 1){     System.out.println(kStartNode.data+" ");     return Boolean.TRUE;    }    boolean kright =kcmLevelOrderTraversalRightToLeft(kStartNode.kright, klevel-1);    boolean kleft =kcmLevelOrderTraversalRightToLeft(kStartNode.kleft, klevel-1);    return kright || kleft;   }   public static void kcmLevelOrderTraversalRightToLeft(TreeElementNode kStartNode) {    if(null == kStartNode){     return;    }    int level=1;    while(kcmLevelOrderTraversalRightToLeft(kStartNode, level++)){     System.out.println("-&&&&&&&--");    }   }    public static void main(String[] args)  {   // Creating a construct a binary tree   TreeElementNode rootNode=constructBinaryTree();   System.out.println("The level order traversal data output of a binary tree will be from left to right. -->:");   kcmLevelOrderTraversalLeftToRight(rootNode);   System.out.println("The level order traversal data output of a binary tree will be from right to left.-->:");   kcmLevelOrderTraversalRightToLeft(rootNode);  }    public static TreeNode constructBinaryTree()  {   TreeElementNode kk=new TreeElementNode(340);   TreeElementNode node320=new TreeElementNode(320);   TreeElementNode node310=new TreeElementNode(310);   TreeElementNode node330=new TreeElementNode(330);   TreeElementNode node360=new TreeElementNode(360);   TreeElementNode node350=new TreeElementNode(350);   TreeElementNode node370=new TreeElementNode(370);   kk.kleft=node320;   kk.kright=node360;   node320.kleft=node310;   node320.kright=node330;     node360.kleft=node350;   node360.kright=node370;     return kk;  } }

 

 

It features two traversal approaches: level-order traversal carried out from left to right and level-order traversal conducted or undertaken from right to left. Below is an explanation of the code.

Explanation:

1. TreeNode Class:
• A static inner class TreeNode represents a node in the binary tree. It has three fields:
• int data: Stores the value of the node.
• TreeNode left: Reference to the left child node.
• TreeNode right: Reference to the right child node.
• The constructor TreeNode(int data) initializes the node with the provided value.
2. Level Order Traversal from Left to Right:
• levelOrderTraversalLeftToRight(TreeNode startNode, int level):
• This method recursively traverses the tree level by level, starting from the given startNode.
• If the level is 1, it prints the node's data.
• It recursively calls itself for the left and right subtrees, reducing the level by 1 in each recursive call.
• It returns true if the traversal is successful, or false if the startNode is null.
• levelOrderTraversalLeftToRight(TreeNode startNode):
• This method initiates the level-order traversal by calling the previous method with increasing levels until no nodes are left to traverse.
3. Level Order Traversal from Right to Left:
• levelOrderTraversalRightToLeft(TreeNode startNode, int level):
• Similar to the left-to-right traversal method, but the order of traversal is reversed.
• It starts with the right subtree first, followed by the left subtree.
• levelOrderTraversalRightToLeft(TreeNode startNode):
• This method initiates the right-to-left traversal by calling the previous method with increasing levels until no nodes are left to traverse.
4. Main Method:
• The main method creates a binary tree using the createBinaryTree() method.
• It then prints the tree in level order from left to right and right to left using the corresponding traversal methods.
5. Binary Tree Structure:
• The createBinaryTree() method builds a binary tree with the following structure:

markdown

 

140       /  \    120   160    /  \   /  \  110  130  150  170

 

• This tree is used as the input for the traversal methods.

Output:

When you run the code, the output will be:

text

Level Order traversal of binary tree will be left to Right -->: 140 120 160 110 130 150 170 Level Order traversal of binary tree will be Right to Left -->: 140 160 120 170 150 130 110

 

Summary:

• Left-to-right traversal prints nodes level by level starting from the leftmost node on each level.
• Right-to-left traversal prints nodes level by level starting from the rightmost node on each level.

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