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Statistical consistency in phylogenetics has traditionally referred to theaccuracy of estimating phylogenetic parameters for a fixed number of species aswe increase the number of characters. However, as sequences are often of fixedlength (e.g. for a gene) although we are often able to sample more taxa, it isuseful to consider a dual type of statistical consistency where we increase thenumber of species, rather than characters. This raises some basic questions:what can we learn about the evolutionary process as we increase the number ofspecies? In particular, does having more species allow us to infer theancestral state of characters accurately? This question is particularlyrelevant when sequence site evolution varies in a complex way from character tocharacter, as well as for reconstructing ancestral sequences. In this paper, weassemble a collection of results to analyse various approaches for inferringancestral information with increasing accuracy as the number of taxa increases.

Inferring ancestral sequences in taxon-rich phylogenies