# Diameter of a Binary Indexed Tree with N nodes

Given a Binary Indexed Tree with N nodes except root node 0 (Numbered from 1 to N), Find its diameter.

Binary Indexed Tree is a tree where parent of a node number X = X – (X & (X – 1)) i.e. last bit is unset in X. The diameter of a tree is the longest simple path between any two leaves.

**Examples:**

Input:N = 12

Output:6

Explanation:Path from node 7 to node 11.

Input : n = 15

Output : 7

**Approach:**

- In a BIT, root is always node 0. In first level, all nodes are of power of 2 . (1, 2, 4, 8, ….)
- Consider any node in the first level (1, 2, 4, 8, ) its sub-tree will include all the nodes which has same number of bits as that of the root.
- Sub-Tree with root 1 will have no child.
- Sub-Tree with root 2 will have 3 as a child.
- Sub-Tree with root 4 will have 5, 6, 7 as a child.
- Sub-Tree with root 8 will have 9, 10, 11, 12, 13, 14, 15 as a child. (Double the size of the previous subtree)
- So subtree with root K will have K nodes including root. And the height of each subtree would be equal:
- for subtree with root 1
- for subtree with root 2
- for subtree with root 4

- Now, we need to find the subtree in which N lies. Say, the height of subtree just before the subtree in which N lies is H and size is L. So, the following cases are possible :
**Case 1 :**When N >= L*2 – 1, in such a scenario N is in last level of its subtree. Thus, the diameter will be 2*H + 1. (Path from the lowest level leaf of the previous subtree to the N ).**Case 2 :**When N >= L + L/2 – 1, in such a scenario N is at level H in its subtree. Thus, diameter will be 2*H.**Case 3 :**Otherwise, it is optimal to consider the maximum path length between leaf nodes of two subtree just before the subtree in which N lies i.e diameter is 2*H – 1.

Below are the implementation of the above approach:

## C++

`#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function to find diameter ` `// of BIT with N + 1 nodes ` `int` `diameter(` `int` `n) ` `{ ` ` ` `// L is size of subtree just before subtree ` ` ` `// in which N lies ` ` ` `int` `L, H, templen; ` ` ` `L = 1; ` ` ` ` ` `// H is the height of subtree just before ` ` ` `// subtree in which N lies ` ` ` `H = 0; ` ` ` ` ` `// Base Cases ` ` ` `if` `(n == 1) { ` ` ` `return` `1; ` ` ` `} ` ` ` `if` `(n == 2) { ` ` ` `return` `2; ` ` ` `} ` ` ` `if` `(n == 3) { ` ` ` `return` `3; ` ` ` `} ` ` ` ` ` `// Size of subtree are power of 2 ` ` ` `while` `(L * 2 <= n) { ` ` ` `L *= 2; ` ` ` `H++; ` ` ` `} ` ` ` ` ` `// 3 Cases as explained in Approach ` ` ` `if` `(n >= L * 2 - 1) ` ` ` `return` `2 * H + 1; ` ` ` `else` `if` `(n >= L + (L / 2) - 1) ` ` ` `return` `2 * H; ` ` ` `return` `2 * H - 1; ` `} ` ` ` `// Driver Code ` `int` `main() ` `{ ` ` ` `int` `n = 15; ` ` ` `cout << diameter(n) << endl; ` `} ` |

*chevron_right*

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## Java

`// Java implementation of the approach ` `class` `GFG ` `{ ` ` ` `// Function to find diameter ` `// of BIT with N + 1 nodes ` `static` `int` `diameter(` `int` `n) ` `{ ` ` ` `// L is size of subtree just before subtree ` ` ` `// in which N lies ` ` ` `int` `L, H, templen; ` ` ` `L = ` `1` `; ` ` ` ` ` `// H is the height of subtree just before ` ` ` `// subtree in which N lies ` ` ` `H = ` `0` `; ` ` ` ` ` `// Base Cases ` ` ` `if` `(n == ` `1` `) { ` ` ` `return` `1` `; ` ` ` `} ` ` ` `if` `(n == ` `2` `) { ` ` ` `return` `2` `; ` ` ` `} ` ` ` `if` `(n == ` `3` `) { ` ` ` `return` `3` `; ` ` ` `} ` ` ` ` ` `// Size of subtree are power of 2 ` ` ` `while` `(L * ` `2` `<= n) { ` ` ` `L *= ` `2` `; ` ` ` `H++; ` ` ` `} ` ` ` ` ` `// 3 Cases as explained in Approach ` ` ` `if` `(n >= L * ` `2` `- ` `1` `) ` ` ` `return` `2` `* H + ` `1` `; ` ` ` `else` `if` `(n >= L + (L / ` `2` `) - ` `1` `) ` ` ` `return` `2` `* H; ` ` ` `return` `2` `* H - ` `1` `; ` `} ` ` ` `// Driver Code ` `public` `static` `void` `main(String []args) ` `{ ` ` ` `int` `n = ` `15` `; ` ` ` ` ` `System.out.println(diameter(n)); ` `} ` `} ` ` ` `// This code contributed by PrinciRaj1992 ` |

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## Python3

`# Python3 implementation of the approach ` ` ` `# Function to find diameter ` `# of BIT with N + 1 nodes ` `def` `diameter(n): ` ` ` ` ` `# L is size of subtree just before ` ` ` `# subtree in which N lies ` ` ` `L, H, templen ` `=` `0` `, ` `0` `, ` `0` `; ` ` ` `L ` `=` `1` `; ` ` ` ` ` `# H is the height of subtree just before ` ` ` `# subtree in which N lies ` ` ` `H ` `=` `0` `; ` ` ` ` ` `# Base Cases ` ` ` `if` `(n ` `=` `=` `1` `): ` ` ` `return` `1` `; ` ` ` ` ` `if` `(n ` `=` `=` `2` `): ` ` ` `return` `2` `; ` ` ` ` ` `if` `(n ` `=` `=` `3` `): ` ` ` `return` `3` `; ` ` ` ` ` `# Size of subtree are power of 2 ` ` ` `while` `(L ` `*` `2` `<` `=` `n): ` ` ` `L ` `*` `=` `2` `; ` ` ` `H ` `+` `=` `1` `; ` ` ` ` ` `# 3 Cases as explained in Approach ` ` ` `if` `(n >` `=` `L ` `*` `2` `-` `1` `): ` ` ` `return` `2` `*` `H ` `+` `1` `; ` ` ` `elif` `(n >` `=` `L ` `+` `(L ` `/` `2` `) ` `-` `1` `): ` ` ` `return` `2` `*` `H; ` ` ` `return` `2` `*` `H ` `-` `1` `; ` ` ` `# Driver Code ` `n ` `=` `15` `; ` `print` `(diameter(n)); ` ` ` `# This code is contributed by Rajput-Ji ` |

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## C#

`// C# implementation of the approach ` `using` `System; ` ` ` `class` `GFG ` `{ ` ` ` `// Function to find diameter ` `// of BIT with N + 1 nodes ` `static` `int` `diameter(` `int` `n) ` `{ ` ` ` `// L is size of subtree just before subtree ` ` ` `// in which N lies ` ` ` `int` `L, H; ` ` ` `L = 1; ` ` ` ` ` `// H is the height of subtree just before ` ` ` `// subtree in which N lies ` ` ` `H = 0; ` ` ` ` ` `// Base Cases ` ` ` `if` `(n == 1) ` ` ` `{ ` ` ` `return` `1; ` ` ` `} ` ` ` `if` `(n == 2) ` ` ` `{ ` ` ` `return` `2; ` ` ` `} ` ` ` `if` `(n == 3) ` ` ` `{ ` ` ` `return` `3; ` ` ` `} ` ` ` ` ` `// Size of subtree are power of 2 ` ` ` `while` `(L * 2 <= n) ` ` ` `{ ` ` ` `L *= 2; ` ` ` `H++; ` ` ` `} ` ` ` ` ` `// 3 Cases as explained in Approach ` ` ` `if` `(n >= L * 2 - 1) ` ` ` `return` `2 * H + 1; ` ` ` `else` `if` `(n >= L + (L / 2) - 1) ` ` ` `return` `2 * H; ` ` ` `return` `2 * H - 1; ` `} ` ` ` `// Driver Code ` `public` `static` `void` `Main(String []args) ` `{ ` ` ` `int` `n = 15; ` ` ` ` ` `Console.WriteLine(diameter(n)); ` `} ` `} ` ` ` `// This code is contributed by 29AjayKumar ` |

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**Output:**

7

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- Finding the lexicographically smallest diameter in a binary tree
- Binary Indexed Tree : Range Updates and Point Queries
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- Sum of all leaf nodes of binary tree
- XOR of path between any two nodes in a Binary Tree
- Product of all nodes in a Binary Tree
- Sum of all nodes at Kth level in a Binary Tree
- Sum of all the Boundary Nodes of a Binary Tree

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