Abstract: Dendritic tree morphology of a neuron is basically a finite, compact metric tree graph. We extend neuronal cable theory to this type of graph domain, and attack a number of forward and inverse problems. For inverse problems, given certain types of (biologically relevant) boundary measurements, we have concentrated on parameter identification problems, where the parameter is spatially varying over an interval (single edge), or over the graph. These parameters have been conductances, or single channel densities. I will mention a couple different numerical approaches, and a recent boundary control approach. For forward problems, we have formulated energy and comparison principle methods in order to examine nonlinear threshold and conduction properties for use on graph domains.