  A tree $T$ on $2^n$ vertices is called set-sequential if the elements in
$V(T)\cup E(T)$ can be labeled with distinct nonzero $(n+1)$-dimensional
$01$-vectors such that the vector labeling each edge is the component-wise sum
modulo $2$ of the labels of the endpoints. It has been conjectured that all
trees on $2^n$ vertices with only odd degree are set-sequential (the "Odd Tree
Conjecture"), and in this paper, we present progress toward that conjecture. We
show that certain kinds of caterpillars (with restrictions on the degrees of
the vertices, but no restrictions on the diameter) are set-sequential.
Additionally, we introduce some constructions of new set-sequential graphs from
smaller set-sequential bipartite graphs (not necessarily odd trees). We also
make a conjecture about pairings of the elements of $\mathbb{F}_2^n$ in a
particular way; in the process, we provide a substantial clarification of a
proof of a theorem that partitions $\mathbb{F}_2^n$ from a 2011 paper by
Balister et al. Finally, we put forward a result on bipartite graphs that is a
modification of a theorem in Balister et al.
