AbstractWe investigate island systems with continuous height functions and strongly laminar systems which are laminar systems containing sets with disjoint boundaries. In the discrete case, we show that for a maximal rectangular system of islands$ \mathcal{H} $on an$m$by$n$rectangular grid we have$\lceil \min (m, n)/ 4\rceil \leq \vert \mathcal{H} \vert \leq \lceil m/ 2\rceil \lceil n/ 2\rceil $. In the continuous case we show that under some conditions maximal strongly laminar systems$ \mathcal{H} $have cardinality${\aleph }_{0} $or${2}^{{\aleph }_{0} } $and present examples with$\vert \mathcal{H} \vert = {\aleph }_{0} $.