ct.category theory – Trees in chain complexes


$DeclareMathOperator{Ch}{mathit{Ch}}$Let $Ch_mathbb{Q}$ denotes the model category of chain complexes over rational numbers. Let $T_ast$ be a tree in $Ch_{mathbb{Q}}$ with $n$ vertices.

How to classify trees with respect to weak equivalences i.e., chain homotopies? Is it true that the classification can be recovered from the $mathit{ho}(Ch_{mathbb{Q}})$?

I think the key factor is that any chain complex $C_astcong oplus_n Vlangle nrangle_ast$, here $Vlangle nrangle_ast$ is the chain complex concentrated in degree $n$ and $Vlangle nrangle_n= H_n(C_ast)$

For example if we take a path with length 2, $f_ast : C_ast to C_ast’$ then it is equivalent to maps $H_n(f_ast): H_n(C_ast) to H_n(C_ast’)$, that is maps between vector spaces. We know that any map between vector spaces is completely describe by the dim(Ker). In this case, any path of length 2 is fully describe by $dim(mathrm{ker}(H_n(f_ast)))_n$.

I really appreciate it if someone could say something for trees.