  Strong Disorder Renormalization for the Random Transverse Field Ising model
leads to a complicated topology of surviving clusters as soon as $d>1$. Even if
one starts from a Cayley tree, the network of surviving renormalized clusters
will contain loops, so that no analytical solution can been obtained. Here we
introduce a modified procedure called 'Boundary Strong Disorder
Renormalization' that preserves the tree structure, so that one can write
simple recursions with respect to the number of generations. We first show that
this modified procedure allows to recover exactly most of the critical
exponents for the one-dimensional chain. After this important check, we study
the RG equations for the quantum Ising model on a Cayley tree with a uniform
ferromagnetic coupling $J$ and random transverse fields with support
$[h_{min},h_{max}]$. We find the following picture (i) for $J>h_{max}$, only
bonds are decimated, so that the whole tree is a quantum ferromagnetic cluster
(ii) for $J<h_{min}$, only sites are decimated, so that no quantum
ferromagnetic cluster is formed, and the ferromagnetic coupling to the boundary
coincides with the partition function of a Directed Polymer model in a random
medium (iii) for $h_{min}<J<h_{max}$, both sites and bonds can be decimated :
the quantum ferromagnetic clusters can either remain finite (the physics is
then similar to (ii), with a quantitative mapping to a modified Directed
Polymer model) or an infinite quantum ferromagnetic cluster appears. We find
that the quantum transition can be of two types : (a) either the quantum
transition takes place in the region where quantum ferromagnetic clusters
remain finite, and the singularity of the ferromagnetic coupling to the
boundary involves the typical correlation length exponent $\nu_{typ}=1$ (b) or
the quantum transition takes place at the point where an extensive quantum
ferromagnetic cluster appears.
