A new approach to counting broadcast trees

In this paper, we derive the formula to count broadcast trees based on the number of internal nodes in the tree. To the best of our knowledge, it is the first time that tree enumeration formulas based on the number of internal nodes in the tree have been explored and addressed. In addition, we give a new proof of Cayley´s famous formula for counting labeled trees. A distinct feature in this proof is that we purely use combinatorial structures instead of constructing a bijection between two kinds of labeled trees as almost all of the existing proofs do by Cameron, P.J., et al, (1994).