Let $Q$ be a quiver which is a connected tree and let $A=KQ/I$ be a quiver algebra with $I$ an admissible ideal, meaning that $I$ is generated by paths of length $\geq 2$. Let $n$ be the number of vertices of $Q$. The Cartan matrix $C_A$ of $A$ is defined as the $n \times n$-matrix with entries $c_{i,j}$ given by the number of non-zero paths in $A$ from $j$ to $i$.
If it helps we can assume that $Q$ is simply laced for the following questions.
Question 1: Is there an easy way to see whether two such algebras $A_1=KQ/I_1$ and $A_2=KQ/I_2$ for the same quiver but possibly different ideals $I_i$ are isomorphic as $K$-algebras?
Im especially interested in a way to do this with GAP and its package QPA. In general testing for isomorphism for quivers is probably not possible in a quick manner, but since the quivers are trees and the relations monomial maybe there is a trick to reduce this problem even to some elementary linear algebra depending on the Cartan matrices of $A$.
Question 2: Is there a combinatorial formula for the number of tree quiver algebras for a fixed $Q$ (at least explicit enough for a computer to calculate the number quickly in linear time)?
A nice example is when $Q$ is a linear oriented line, where we obtain the Catalan numbers.
Probably we cant expect a nice answer to the next question, but sometimes combinatorics is surprising:
Question 3: Is there a nice closed formula for the number of all simply laced tree quiver algebras having $n$ vertices?
