  This work is concerned with tree tensor network operators (TTNOs) for
representing quantum Hamiltonians. We first establish a mathematical framework
connecting tree topologies with state diagrams. Based on these, we devise an
algorithm for constructing a TTNO given a Hamiltonian. The algorithm exploits
the tensor product structure of the Hamiltonian to add paths to a state
diagram, while combining local operators if possible. We test the capabilities
of our algorithm on random Hamiltonians for a given tree structure.
Additionally, we construct explicit TTNOs for nearest neighbour interactions on
a tree topology. Furthermore, we derive a bound on the bond dimension of tensor
operators representing arbitrary interactions on trees. Finally, we consider an
open quantum system in the form of a Heisenberg spin chain coupled to bosonic
bath sites as a concrete example. We find that tree structures allow for lower
bond dimensions of the Hamiltonian tensor network representation compared to a
matrix product operator structure. This reduction is large enough to reduce the
number of total tensor elements required as soon as the number of baths per
spin reaches $3$.
