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In all cases, very large input trees (10,000 taxa) can easily be processed and transformed into time-scaled trees.

We compare these algorithms to standard methods (root-to-tip, r8s version of Langley–Fitch method, and BEAST).

Our algorithms exploit the tree (recursive) structure of the problem at hand, and the close relationships between least-squares and linear algebra.

We distinguish between an unconstrained setting and the case where the temporal precedence constraint (i.e., an ancestral node must be older that its daughter nodes) is accounted for.

Phylogenies provide a useful way to understand the evolutionary history of genetic samples, and data sets with more than a thousand taxa are becoming increasingly common, notably with viruses (e.g., human immunodeficiency virus (HIV)).

Dating ancestral events is one of the first, essential goals with such data.

An Online Appendix, providing additional algorithm descriptions, tables, and figures can be found in the Supplementary Material available on Dryad at .