What is a Infinetly Rooted Tree?

I was trying to solve the problem k-trees,
Actual Problem Link :- Problem - 431C - Codeforces
and was stuck to find how given test cases are giving answer , if anyone could help me with it! That would be really great.

Infinitely Rooted Tree means that the Tree has infinite number of node with a designated root.
As far as the explanation of the test cases:
1.) In the 1st test-case ( n=3,k=3,d=2)
Ans:- 3 ( [1,2], [2,1], [3] )
2.) In the 2nd test-case (n=3,k=3,d=3)
Ans:- 1 ( [3] )
As far as solution is considered. Question is same as finding Number of different ways to reach a score of N with possible length of each step should be less than or equal to k with at least 1 step length should be greater than or equal to d.

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