  A long-standing conjecture on spanning trees of a hypercube states that a
balanced tree on $2^n$ vertices with maximum degree at most $3$ spans the
hypercube of dimension $n$ \cite{havel1986}. In this paper, we settle the
conjecture for a special family of binary trees. A $0$-caterpillar is a path.
For $k\geq 1$, a $k$-caterpillar is a binary tree consisting of a path with
$j$-caterpillars $(0\leq j\leq k-1)$ emanating from some of the vertices on the
path. A $k$-caterpillar that contains a perfect matching is said to be
perfectly balanced. In this paper, we show that a perfectly balanced
$2$-caterpillar on $2^n$ vertices spans the hypercube of dimension $n$.
