Genomic rearrangement operations can be very useful to infer the phylogenetic relationship of gene orders representing species. We study the problem of finding potential ancestral gene orders for the gene orders of given taxa, such that the corresponding rearrangement scenario has a minimal number of reversals, and where each of the reversals has to preserve the common intervals of the given input gene orders. Common intervals identify sets of genes that occur consecutively in all input gene orders. The problem of finding such an ancestral gene order is called the preserving reversal median problem (pRMP). A tree-based data structure for the representation of the common intervals of all input gene orders is used in our exact algorithm TCIP for solving the pRMP. It is known that the minimum number of reversals to transform one gene order into another can be computed in polynomial time, whereas the corresponding problem with the restriction that common intervals should not be destroyed is already NP-hard. It is shown theoretically that TCIP can solve a large class of pRMP instances in polynomial time. Empirically we show the good performance of TCIP on biological and artificial data.