  Ancestral state reconstruction is one of the most important tasks in
evolutionary biology. Conditions under which we can reliably reconstruct the
ancestral state have been studied for both discrete and continuous traits.
However, the connection between these results is unclear, and it seems that
each model needs different conditions. In this work, we provide a unifying
theory on the consistency of ancestral state reconstruction for various types
of trait evolution models. Notably, we show that for a sequence of nested trees
with bounded heights, the necessary and sufficient conditions for the existence
of a consistent ancestral state reconstruction method under discrete models,
the Brownian motion model, and the threshold model are equivalent. When tree
heights are unbounded, we provide a simple counter-example to show that this
equivalence is no longer valid.
