We provide necessary and sufficient conditions on the unimodality of a
convolution of two sequences of binomial coefficients preceded by a finite
number of ones. These convolution sequences arise as as rank sequences of
posets of vertex-induced subtrees for a particular class of trees. The number
of such trees whose poset of vertex-induced subgraphs containing the root is
not rank unimodal is determined for a fixed number of vertices $i$