  We study infinite tree and ultrametric matrices, and their action on the
boundary of the tree. For each tree matrix we show the existence of a symmetric
random walk associated to it and we study its Green potential. We provide a
representation theorem for harmonic functions that includes simple expressions
for any increasing harmonic function and the Martin kernel. In the boundary, we
construct the Markov kernel whose Green function is the extension of the matrix
and we simulate it by using a cascade of killing independent exponential random
variables and conditionally independent uniform variables. For ultrametric
matrices we supply probabilistic conditions to study its potential properties
when immersed in its minimal tree matrix extension.
