On the finiteness of prime trees and their relation to modular forms

In this paper, we introduce the prime trees associated with a finite subset $P$ of the set of all prime numbers, and provide conditions under which the tree is of finite type. Moreover, we compute the density of finite-type subsets $P$. As an application, we show that for weight $k\ge 2$ and levels $N=N^{\mathrm{\prime }}{\prod }_{p\in P}{p}^{a_p}$, where $N^{\mathrm{\prime }}$ is squarefree and $a_p\ge 2$, every cusp form $f\in {\mathcal{S}}_k({\mathrm{\Gamma }}_0(N))$ can be expressed as a linear combination of products of two specific Eisenstein series whenever $P$ is of finite type.