  An erasure code is said to be a code with sequential recovery with parameters
$r$ and $t$, if for any $s \leq t$ erased code symbols, there is an $s$-step
recovery process in which at each step we recover exactly one erased code
symbol by contacting at most $r$ other code symbols. In earlier work by the
same authors, presented at ISIT 2017, we had given a construction for binary
codes with sequential recovery from $t$ erasures, with locality parameter $r$,
which were optimal in terms of code rate for given $r,t$, but where the block
length was large, on the order of $r^{c^t}$, for some constant $c >1$. In the
present paper, we present an alternative construction of a rate-optimal code
for any value of $t$ and any $r\geq3$, where the block length is significantly
smaller, on the order of $r^{\frac{5t}{4}+\frac{7}{4}}$ (in some instances of
order $r^{\frac{3t}{2}+2}$). Our construction is based on the construction of
certain kind of tree-like graphs with girth $t+1$. We construct these graphs
and hence the codes recursively.
