  A theorem of Strichartz states that if a uniformly bounded bi-infinite
sequence of functions on the Euclidean spaces, satisfies the condition that the
Laplacian acting on a function in this sequence yields the next one, then each
function in this sequence is an eigenfunction of the Laplacian. We consider a
generalization of this result for homogeneous trees, where we replace bounded
functions by tempered distributions and the Laplacian by multiplier operators
acting on the tempered distributions. After establishing the result in this
general context, we narrow our focus to specific cases, which includes
important examples of multiplier operators such as heat and Schr\"odinger
operator, ball and sphere averages of function.
