Let $T$ be a tree (an acyclic connected graph) on the vertex set $[n]=\{1,\dots, n\}$.
Let $A$ be the adjacency matrix of $T$, i.e., the $n\times n$ matrix with $A_{ij} = 1$ if $i$ and $j$ are adjacent in $T$ and $A_{ij}=0$ otherwise. Prove that the number of nonnegative eigenvalues of $A$ equals to the size of the largest independent set of $T$. Here, an independent set is a set of vertices where no two vertices in the set are adjacent.