Abstract— Algorithms to reconstruct character evolution on polytomous cladograms or phylogenetic trees have to date interpreted each polytomy literally, as if it were an event of multiple speciation, with multiple daughter species descending independently from a mother species, thus requiring any similarities shared by only some of these daughters to be accounted for by convergence. These algorithms are not appropriate when the polytomy is interpreted in the usual way, namely as representing uncertainty in the cladogram's resolution. New algorithms for both ordered and unordered characters are presented to reconstruct character evolution under the uncertain-resolution interpretation of polytomies. These algorithms allow the cladogram to resolve itself so as to be favourable for the character whose evolution is being reconstructed. Because different characters may have different favourable resolutions, it is not possible in general to use these algorithms to determine the total parsimony of a polytomous cladogram (the number of evolutionary steps required over all characters by the cladogram), for which the only adequate approach is to find a most parsimonious dichotomous resolution of the cladogram.