The paper addresses the selection of the best representations for distributed and/or dependent signals. Given an indexed tree structured library of bases and a semi-collaborative distribution scheme associated with minimum information exchange (emission and reception of one single index corresponding to a marginal best basis), the paper proposes the median basis computed on a set of best marginal bases for joint representation or fusion of distributed/dependent signals. The paper provides algorithms for computing this median basis with respect to standard tree structured libraries of bases such as wavelet packet bases or cosine trees. These algorithms are effective when an additive information cost is under consideration. Experimental results performed on distributed signal compression confirms worthwhile properties for the median of marginal best bases with respect to the ideal best joint basis, the latter being underdetermined in practice, except when a full collaboration scheme is under consideration.