not contain any backstep, where a backstep is a step which increases the distance to the midpoint. Thus we only consider steps which are either horizontal or upwards in Fig. \[fig:latt\](b). Also note that if $s_{l}=0$, the route goes directly to the closest point on level $l-1$. Thus a complete sequence of $l$ route-marker labels $(l,d_{l},s_{l})$ gives a class of routes from the surface point to the midpoint.

The next step is to consider how we get from the starting point on level $l$ to the end point on level $l-1$ for one specific link configuration. As one moves horizontally on level $l$, one arrives at the possible route-markers in consecutive order. Some of the lattice points are linked to the level above, while others are not. Only the ones which are linked are possible route-markers. These possible route-marker points can be enumerated in consecutive order $s_{l}^{(k)}$ with $k=1,2,...$, so that $(l,d_{l},s_{l}^{(k)})$ enumerates all the possible route classes we can have when starting from the first point on level $l$ going horizontally in the direction $d_{l}$ to the next route marker, which connects to the level $l-1$. Out of these possibly many classes one picks one specific, the one with 