The number of trees whose sum of vertex degrees is L

The number of trees for a given integer degree L can be determined by an equation based on the graph assumption. First, we note that the integer degree of a tree with N vertices is continuously 2N-2. Using this, we can calculate the number of clearances in the tree, which is L minus 2. Another method is to determine the number of internal vertices by subtracting the number of takeoffs from the total number of vertices. Finally, we can determine the number of ways to distribute the remaining degrees among interior vertices by using a combination equation. Therefore, these steps can be used to count the number of trees of integer degree L.

usage instructions

• Recursive enumeration

• Combined analysis

Recursive enumeration

Recursive counting can be a strategy to determine the number of trees with a given degree L. Starting from a single vertex, we systematically include modern vertices and edges to build the tree. At each step, we pass the remaining degrees between existing vertices while keeping the specified sum. This process is repeated recursively, exploring all possible combinations. By backtracking and considering different degrees, we are able to determine the number of valid trees. This approach is very useful for smaller input sizes, but may become inefficient for larger values ​​of L since its time complexity is exponential.

algorithm

• Define a recursive function countTrees(L, vertexCount), where L is the required sum of degrees and vertexCount represents the number of vertices in the tree.

• basic situation.

• If L is balanced with 2 and vertexCount equals 1, return 1 (because a single vertex can be a substantial tree). Initialize the variable treeCount to 0.

• The possible degrees of traversing the current vertex from 1 to L-1.

• Inside the loop.

• Subtract L from the current degree and call countTrees recursively with the upgraded L and vertexCount-1. Assign the returned value to treeCount. Return treeCount.

• Call count. Trees work with the required L and starting vertex count (according to rule 1) to get the number of trees with a given sum of degrees.

Example

Example

#include <iostream>

int countTrees(int L, int vertexCount) {
   if (L == 2 && vertexCount == 1) {
     return 1;
   }

   int treeCount = 0;
   for (int degree = 1; degree <= L - 1; degree++) {
      int remainingSum = L - degree;
      int subTreeCount = countTrees(remainingSum, vertexCount - 1);
      treeCount += subTreeCount;
   }

   return treeCount;
}

int main() {
   int L = 8;
   int vertexCount = 4;

   int numTrees = countTrees(L, vertexCount);
   std::cout << "Number of trees with sum of degrees " << L << " and " << vertexCount << " vertices: " << numTrees << std::endl;

   return 0;
}

Output

Number of trees with sum of degrees 8 and 4 vertices: 10

Combined analysis

Combinatorial analysis is the process of considering the structure of the trees and numbering them using a combination strategy when determining the number of trees with a given degree sum L. It involves analyzing the degree requirements, determining the number of internal vertices and clearing them, and assigning the remaining degrees to them. By exploiting concepts such as combinations, stages, and recurring relationships, combinatorial analysis allows deriving equations or computational methods to accurately count the number of trees with a specific degree sum L. This approach utilizes scientific standards to solve problems efficiently and skillfully.

algorithm

• Initialize the variable count_trees to 0.

• Iterate the number of possible filters from 1 to L-2, i.

• Calculate the number of internal vertices, that is, L - i.

• Use the combination method to assign the remaining degrees to interior vertices. You can compute this distribution using recursive methods or dynamic programming.

• Increase count_trees to the number of items by passing degrees between internal vertices and selecting leaves.

• Return count_trees as the result.

Example

Example

#include <iostream>
#include <algorithm>
#include <vector>

int countTrees(int L) {
   int count_trees = 0;
   for (int i = 1; i <= L - 2; i++) {
      int internal_vertices = L - i;
        
      // Calculate the number of ways to distribute degrees among internal vertices
      std::vector<int> degrees(internal_vertices - 2, 1);
      degrees.push_back(2);
        
      do {
         // Calculate the number of ways to select the leaves
         int leaf_selection = std::count_if(degrees.begin(), degrees.end(), [](int x) {
            return x == 1;
         });
            
         count_trees += leaf_selection;
      } while (std::next_permutation(degrees.begin(), degrees.end()));
   }

   return count_trees;
}

int main() {
   int L = 10; // Assuming L = 10
   int result = countTrees(L);

   std::cout << "Number of trees: " << result << std::endl;

   return 0;
}

Output

Number of trees: 168

in conclusion

This article explores two strategies for determining the number of trees with a specific degree in a graph. The main strategy is recursive identification, which creates the tree by efficiently adding vertices and edges, passing along the remaining degrees. The second strategy is combinatorial analysis, which utilizes numerical criteria and combinatorial methods to compute the number of trees by passing degrees between vertices. Both methods provide efficient ways to solve problems and are relevant in completely different scenarios.

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