The vertex covers, Betti numbers and projective dimensions of perfect binary trees

Let $T$ be a perfect binary tree and $I$ be its edge ideal in the polynomial ring $S$. We determine the vertex cover number, independent number, and establish the recursive formula to compute the number of minimal vertex covers. As a consequence, we compute the depth and projective dimension of $S\mathrm{/}I$ and show that the total Betti number of $S\mathrm{/}I$ at the highest homological degree always equals one.