Diameter of a Binary Tree

Binary Tree

Easy

Given a binary tree, you need to compute the length of the diameter of the tree. The diameter of a binary tree is the length of the longest path between any two nodes in a tree. This path may or may not pass through the root.

Example:
Given a binary tree

          1
         / \
        2   3
       / \     
      4   5

Return3, which is the length of the path [4,2,1,3] or [5,2,1,3].

Note:The length of path between two nodes is represented by the number of edges between them.


Source: https://www.geeksforgeeks.org/diameter-of-a-binary-tree/

The diameter of a tree (sometimes called the width) is the number of nodes on the longest path between two end nodes.

The diagram below shows two trees each with diameter nine, the leaves that form the ends of a longest path are shaded (note that there is more than one path in each tree of length nine, but no path longer than nine nodes).

Solution & Analysis

The diameter of a tree T is the largest of the following quantities:

* the diameter of T’s left subtree
* the diameter of T’s right subtree
* the longest path between leaves that goes through the root of T (this can be computed from the heights of the subtrees of T)

LeetCode

public class Solution {
    int max = 0;

    public int diameterOfBinaryTree(TreeNode root) {
        maxDepth(root);
        return max;
    }

    private int maxDepth(TreeNode root) {
        if (root == null) return 0;

        int left = maxDepth(root.left);
        int right = maxDepth(root.right);

        max = Math.max(max, left + right);

        return Math.max(left, right) + 1;
    }
}

Straightforward implementation:

O(n^2) time complexity

// Recursive optimized Java program to find the diameter of a 
// Binary Tree 

/* Class containing left and right child of current 
node and key value*/
class Node 
{ 
    int data; 
    Node left, right; 

    public Node(int item) 
    { 
        data = item; 
        left = right = null; 
    } 
} 

/* Class to print the Diameter */
class BinaryTree 
{ 
    Node root; 

    /* Method to calculate the diameter and return it to main */
    int diameter(Node root) 
    { 
        /* base case if tree is empty */
        if (root == null) 
            return 0; 

        /* get the height of left and right sub trees */
        int lheight = height(root.left); 
        int rheight = height(root.right); 

        /* get the diameter of left and right subtrees */
        int ldiameter = diameter(root.left); 
        int rdiameter = diameter(root.right); 

        /* Return max of following three 
        1) Diameter of left subtree 
        2) Diameter of right subtree 
        3) Height of left subtree + height of right subtree + 1 */
        return Math.max(lheight + rheight + 1, 
                        Math.max(ldiameter, rdiameter)); 

    } 

    /* A wrapper over diameter(Node root) */
    int diameter() 
    { 
        return diameter(root); 
    } 

    /*The function Compute the "height" of a tree. Height is the 
    number f nodes along the longest path from the root node 
    down to the farthest leaf node.*/
    static int height(Node node) 
    { 
        /* base case tree is empty */
        if (node == null) 
            return 0; 

        /* If tree is not empty then height = 1 + max of left 
        height and right heights */
        return (1 + Math.max(height(node.left), height(node.right))); 
    } 

    public static void main(String args[]) 
    { 
        /* creating a binary tree and entering the nodes */
        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("The diameter of given binary tree is : "
                        + tree.diameter()); 
    } 
}

Optimized implementation:

The above implementation can be optimized by calculating the height in the same recursion rather than calling a height() separately. Thanks to Amar for suggesting this optimized version. This optimization reduces time complexity to O(n).

O(n) time complexity

// Recursive Java program to find the diameter of a 
// Binary Tree 

/* Class containing left and right child of current 
node and key value*/
class Node 
{ 
    int data; 
    Node left, right; 

    public Node(int item) 
    { 
        data = item; 
        left = right = null; 
    } 
} 

// A utility class to pass heigh object 
class Height 
{ 
    int h; 
} 

/* Class to print the Diameter */
class BinaryTree 
{ 
    Node root; 

    /* define height =0 globally and call diameterOpt(root,height) 
    from main */
    int diameterOpt(Node root, Height height) 
    { 
        /* lh --> Height of left subtree 
        rh --> Height of right subtree */
        Height lh = new Height(), rh = new Height(); 

        if (root == null) 
        { 
            height.h = 0; 
            return 0; /* diameter is also 0 */
        } 

        /* ldiameter --> diameter of left subtree 
        rdiameter --> Diameter of right subtree */
        /* Get the heights of left and right subtrees in lh and rh 
        And store the returned values in ldiameter and ldiameter */
        int ldiameter = diameterOpt(root.left, lh); 
        int rdiameter = diameterOpt(root.right, rh); 

        /* Height of current node is max of heights of left and 
        right subtrees plus 1*/
        height.h = Math.max(lh.h, rh.h) + 1; 

        return Math.max(lh.h + rh.h + 1, Math.max(ldiameter, rdiameter)); 
    } 

    /* A wrapper over diameter(Node root) */
    int diameter() 
    { 
        Height height = new Height(); 
        return diameterOpt(root, height); 
    } 

    /*The function Compute the "height" of a tree. Height is the 
    number f nodes along the longest path from the root node 
    down to the farthest leaf node.*/
    static int height(Node node) 
    { 
        /* base case tree is empty */
        if (node == null) 
            return 0; 

        /* If tree is not empty then height = 1 + max of left 
        height and right heights */
        return (1 + Math.max(height(node.left), height(node.right))); 
    } 

    public static void main(String args[]) 
    { 
        /* creating a binary tree and entering the nodes */
        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("The diameter of given binary tree is : "
                        + tree.diameter()); 
    } 
}

Reference

https://www.geeksforgeeks.org/diameter-of-a-binary-tree/

Diameter of a Binary Tree in O(n) [A new method]

Diameter of an N-ary tree

http://www.cs.duke.edu/courses/spring00/cps100/assign/trees/diameter.html

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