  We introduce a dense and a dilute loop model on causal dynamical
triangulations. Both models are characterised by a geometric coupling constant
$g$ and a loop parameter $\alpha$ in such a way that the purely geometric
causal triangulation model is recovered for $\alpha=1$. We show that the dense
loop model can be mapped to a solvable planar tree model, whose partition
function we compute explicitly and use to determine the critical behaviour of
the loop model. The dilute loop model can likewise be mapped to a planar tree
model; however, a closed-form expression for the corresponding partition
function is not obtainable using the standard methods employed in the dense
case. Instead, we derive bounds on the critical coupling $g_c$ and apply
transfer matrix techniques to examine the critical behaviour for $\alpha$
small.
