  We consider the Laplacian on a rooted metric tree graph with branching number
$ K \geq 2 $ and random edge lengths given by independent and identically
distributed bounded variables. Our main result is the stability of the
absolutely continuous spectrum for weak disorder. A useful tool in the
discussion is a function which expresses a directional transmission amplitude
to infinity and forms a generalization of the Weyl-Titchmarsh function to
trees. The proof of the main result rests on upper bounds on the range of
fluctuations of this quantity in the limit of weak disorder.
