Possible edges of a tree for given diameter, height and vertices

• Difficulty Level : Hard
• Last Updated : 14 May, 2021

Find a tree with the given values and print the edges of the tree. Print “-1”, if the tree is not possible.

Given three integers n, d and h.

n -> Number of vertices. [1, n]
d -> Diameter of the tree (largest
distance between two vertices).
h -> Height of the tree (longest distance
between vertex 1 and another vertex)

Examples :

Input : n = 5, d = 3, h = 2
Output : 1 2
2 3
1 4
1 5
Explanation : We can see that the height of the tree is 2 (1 ->
2 --> 5) and diameter is 3 ( 3 -> 2 -> 1 -> 5).
So our conditions are satisfied.

Input :  n = 8, d = 4, h = 2
Output : 1 2
2 3
1 4
4 5
1 6
1 7
1 8
Explanation : 1. Observe that when d = 1, we cannot construct a tree (if tree has more than 2 vertices). Also when d > 2*h, we cannot construct a tree.
2. As we know that height is the longest path from vertex 1 to another vertex. So build that path from vertex 1 by adding edges up to h. Now, if d > h, we should add another path to satisfy diameter from vertex 1, with a length of d – h.
3. Our conditions for height and diameter are satisfied. But still some vertices may be left. Add the remaining vertices at any vertex other than the end points. This step will not alter our diameter and height. Chose vertex 1 to add the remaining vertices (you can chose any).
4. But when d == h, choose vertex 2 to add the remaining vertices.

C++

 // C++ program to construct tree for given count// width and height.#include using namespace std; // Function to construct the treevoid constructTree(int n, int d, int h){    if (d == 1) {         // Special case when d == 2, only one edge        if (n == 2 && h == 1) {            cout << "1 2" << endl;            return;        }        cout << "-1" << endl; // Tree is not possible        return;    }     if (d > 2 * h) {        cout << "-1" << endl;        return;    }     // Satisfy the height condition by add    // edges up to h    for (int i = 1; i <= h; i++)            cout << i << " " << i + 1 << endl;         if (d > h) {         // Add d - h edges from 1 to        // satisfy diameter condition        cout << "1"            << " " << h + 2 << endl;        for (int i = h + 2; i <= d; i++) {            cout << i << " " << i + 1 << endl;        }    }     // Remaining edges at vertex 1 or 2(d == h)    for (int i = d + 1; i < n; i++)    {        int k = 1;        if (d == h)            k = 2;        cout << k << " " << i + 1 << endl;    }} // Driver Codeint main(){    int n = 5, d = 3, h = 2;    constructTree(n, d, h);    return 0;}

Java

 // Java program to construct tree for given count// width and height.class GfG { // Function to construct the treestatic void constructTree(int n, int d, int h){    if (d == 1) {         // Special case when d == 2, only one edge        if (n == 2 && h == 1) {            System.out.println("1 2");            return;        }        System.out.println("-1"); // Tree is not possible        return;    }     if (d > 2 * h) {        System.out.println("-1");        return;    }     // Satisfy the height condition by add    // edges up to h    for (int i = 1; i <= h; i++)            System.out.println(i + " " + (i + 1));         if (d > h) {         // Add d - h edges from 1 to        // satisfy diameter condition        System.out.println("1" + " " + (h + 2));        for (int i = h + 2; i <= d; i++) {            System.out.println(i + " " + (i + 1));        }    }     // Remaining edges at vertex 1 or 2(d == h)    for (int i = d + 1; i < n; i++)    {        int k = 1;        if (d == h)            k = 2;        System.out.println(k + " " + (i + 1));    }} // Driver Codepublic static void main(String[] args){    int n = 5, d = 3, h = 2;    constructTree(n, d, h);}}

Python3

 # Python3 code to construct tree for given count# width and height. # Function to construct the treedef constructTree(n, d, h):    if d == 1:         # Special case when d == 2, only one edge        if n == 2 and h == 1:            print("1 2")            return 0                 print("-1")    # Tree is not possible        return 0         if d > 2 * h:        print("-1")        return 0             # Satisfy the height condition by add    # edges up to h    for i in range(1, h+1):        print(i," " , i + 1)         if d > h:         # Add d - h edges from 1 to        # satisfy diameter condition        print(1,"  ", h + 2)        for i in range(h+2, d+1):            print(i, " " , i + 1)                 # Remaining edges at vertex 1 or 2(d == h)    for i in range(d+1, n):        k = 1        if d == h:            k = 2        print(k ," " , i + 1) # Driver Coden = 5d = 3h = 2constructTree(n, d, h) # This code is contributed by "Sharad_Bhardwaj".

C#

 // C# program to construct tree for // given count width and height.using System; class GfG{     // Function to construct the tree    static void constructTree(int n, int d, int h)    {        if (d == 1)        {             // Special case when d == 2,            // only one edge            if (n == 2 && h == 1)            {                Console.WriteLine("1 2");                return;            }                         // Tree is not possible            Console.WriteLine("-1");            return;        }         if (d > 2 * h)        {            Console.WriteLine("-1");            return;        }         // Satisfy the height condition        // by add edges up to h        for (int i = 1; i <= h; i++)            Console.WriteLine(i + " " + (i + 1));         if (d > h)        {             // Add d - h edges from 1 to            // satisfy diameter condition            Console.WriteLine("1" + " " + (h + 2));            for (int i = h + 2; i <= d; i++)            {                Console.WriteLine(i + " " + (i + 1));            }        }         // Remaining edges at vertex 1 or 2(d == h)        for (int i = d + 1; i < n; i++)        {            int k = 1;            if (d == h)                k = 2;            Console.WriteLine(k + " " + (i + 1));        }    }     // Driver Code    public static void Main(String[] args)    {        int n = 5, d = 3, h = 2;        constructTree(n, d, h);    }} // This code is contributed by 29AjayKumar

Javascript



Output :

1 2
2 3
1 4
1 5

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