  The subnormal completion problem on a directed tree is to determine, given a
collection of weights on a subtree, whether the weights may be completed to the
weights of a subnormal weighted shift on the directed tree. We study this
problem on a directed tree with a single branching point, $\eta$ branches and
the trunk of length $1$ and its subtree which is the "truncation" of the full
tree to vertices of generation not exceeding $2$. We provide necessary and
sufficient conditions written in terms of two parameter sequences for the
existence of a subnormal completion in which the resulting measures are
$2$-atomic. As a consequence, we obtain a solution of the subnormal completion
problem for this pair of directed trees when $\eta < \infty$. If $\eta=2$, we
present a solution written explicitly in terms of initial data.
