Ancestral sequence reconstruction and phylogenetic reconstruction from present-day states are two important and related problems in computational biology. Multiple sequence alignment is a tertiary task that makes important connections between these two problems. Here, we show that perfect sequence alignment with high probability is possible with a dense enough underlying phylogeny influencing the evolution of genetic sequences. New probabilistic techniques in the evolution of sequences with insertions and deletions are developed, with a view toward more general state spaces. Based on work with Louis Fan, Christine Heitsch, and Sebastien Roch.