Abstract. This is the first contribution of a sequence of papers introducing the notions of [Formula: see text]-weak order and [Formula: see text]-permutahedra, certain discrete objects that are indexed by a sequence of nonnegative integers [Formula: see text]. In this first paper, we concentrate purely on the combinatorics and lattice structure of the [Formula: see text]-weak order, a partial order on certain decreasing trees which generalizes the classical weak order on permutations. In particular, we show that the [Formula: see text]-weak order is a semidistributive and congruence uniform lattice, generalizing known results for the classical weak order on permutations. Restricting the [Formula: see text]-weak order to certain trees gives rise to the [Formula: see text]-Tamari lattice, a sublattice which generalizes the classical Tamari lattice. We show that the [Formula: see text]-Tamari lattice can be obtained as a quotient lattice of the [Formula: see text]-weak order when [Formula: see text] has no zeros, and show that the [Formula: see text]-Tamari lattices (for arbitrary [Formula: see text]) are isomorphic to the [Formula: see text]-Tamari lattices of Préville-Ratelle and Viennot. The underlying geometric structure of the [Formula: see text]-weak order will be studied in a sequel of this paper, where we introduce the notions of [Formula: see text]-permutahedra and [Formula: see text]-associahedra.