  We prove that when a class of partial differential equations, generalized
from the cable equation, is defined on tree graphs, and when the inputs are
restricted to a spatially discrete, well chosen set of points, the Green's
function (GF) formalism can be rewritten to scale as $O(n)$ with the number $n$
of input locations, contrary to the previously reported $O(n^2)$ scaling. We
show that the linear scaling can be combined with an expansion of the remaining
kernels as sums of exponentials, to allow efficient simulations of equations
from the aforementioned class. We furthermore validate this simulation paradigm
on models of nerve cells and explore its relation with more traditional finite
difference approaches. Situations in which a gain in computational performance
is expected, are discussed.
