I have read that if $A$ is the adjacency matrix of a tree $T$, then we have that
$\det(\lambda I - A) = \sum_{k=0}^{\lfloor n/2 \rfloor} (-1)^k N_k(T) \lambda^{n-2k} $
where $N_k(T)$ is the number of matchings of size $k$ of $T$.
However, I couldn't find a proof. Does anyone know how to do it? Thanks!