AbstractWe compute the weights, i.e. the values at the minimal idempotents, for the Markov trace on the Hecke algebra of typeBnand typeDn. In order to prove the weight formula, we define representations of the Hecke algebra of typeBonto a reduced Hecke algebra of typeA. To compute the weights for typeDwe use the inclusion of the Hecke algebra of typeDinto the Hecke algebra of typeB.