Mixed finite element methods are capable of imposing physical conservation laws at the discrete level by preserving the structure of governing PDEs. However, this often requires a large number of degrees of freedom and leads to computationally demanding saddle point systems. In this talk, we consider a decomposition of the lowest order finite element spaces based on spanning trees in the grid. This decomposition allows us to derive so-called Poincaré operators for the finite element de Rham complex. Using this abstract tool, we form a new basis in which each (Hodge-)Laplace problem unravels from a large saddle-point problem into four smaller, symmetric positive definite systems. In turn, we achieve a significant computational speed-up, without loss of accuracy, for applications ranging from flow in porous media to electromagnetics and solid mechanics. We proceed by generalizing the Poincaré operator to the mixed-dimensional setting, where the domain is decomposed into subdomains of heterogeneous dimensionality. This allows us to create spanning tree solvers and preconditioners for mixed-dimensional problems. In particular, we showcase the solver for the RT0-P0 discretization of flow in fractured porous media.