Trees and generalised inverse limits on intervals

We consider inverse limits of sequences of upper semicontinuous set-valued functions fi+1:Ii+1→2Ii (where Ii=[0,1] for each i∈N), for which the graph of each bonding function is an arc. We show that any finite tree can be obtained as such an inverse limit, and one for which each bonding function is one of two specified functions. In addition, we discuss trees of height ω+1 that can be obtained as the inverse limit of such a sequence.