Minimising the phonon thermal conductance of self-assembled molecular films, whilst preserving their electrical properties, is highly desirable, both for thermal management at the nanoscale and for the design of high-efficiency thermoelectric materials. Here we highlight a new strategy for minimising the phonon thermal conductance of Christmas-tree-like molecules composed of a long trunk, along which phonons can propagate, attached to pendant molecular branches. We demonstrate that phonon transport along the trunk is suppressed by Fano resonances associated with internal vibrational modes of the branches and that thermal conductance is suppressed most-effectively in molecules with pendant branches of different lengths. As examples, we use density functional theory to demonstrate the reduction in phonon transport in tree-like molecules formed from alkane or acene trunks with various pendant branches.