  In this paper we develop the theory of drift-diffusion on a semi-infinite
Cayley tree with stochastic resetting. In the case of a homogeneous tree with a
closed terminal node and no resetting, it is known that the system undergoes a
classical localization-delocalization (LD) transition at a critical mean
velocity $v_c= -(D/L)\ln (z-1)$ where $D$ is the diffusivity, $L$ is the branch
length and $z$ is the coordination number of the tree. If $v <v_c$ then the
steady state concentration at the terminal node is non-zero (drift-dominated
localized state), whereas it is zero when $v >v_c $ (diffusion-dominated
delocalized state). This is equivalent to the transition between recurrent and
transient transport on the tree, with the mean first passage time (MFPT) to be
absorbed by an open terminal node switching from a finite value to infinity.
Here we show how the LD transition provides a basic framework for understanding
analogous phase transitions in optimal resetting rates. First, we establish the
existence of an optimal resetting rate $r^{**}(z)$ that maximizes the
steady-state solution at a closed terminal node. In addition, we show that
there is a phase transition at a critical velocity $v_c^{**}(z)$ such that
$r^{**}>0$ for $v>v_c^{**}$ and $r^{**}=0$ for $v<v_c^{**}$. We then identify a
critical velocity $v^*(z)$ for a phase transition in a second optimal resetting
rate $r^*$ that minimizes the MFPT to be absorbed by an open terminal node.
Previous results for the semi-infinite line are recovered on setting $z=2$. The
critical velocity of the LD transition provides an upper bound for the other
critical velocities such that $v_c^*(z)<v_c^{**}(z)<v_c(z)$ for all finite $z$.
Only $v_c(z)$ has a simple universal dependence on the coordination number $z$.
We end by considering the combined effects of quenched disorder and stochastic
resetting.
