We develop a diagrammatic calculus for welded and classical knotted objects. We define Arrow presentations, which are essentially equivalent to Gauss diagrams but carry no sign on arrows, and more generally w-tree presentations , which can be seen as 'higher order Gauss diagrams'. We provide a complete set of moves for Arrow and w-tree presentations. This Arrow calculus is used to characterize finite type invariants of welded knots and long knots. Using S. Satoh's Tube map, which realizes welded objects into knotted surfaces in 4-space, we recover several topological results due to K. Habiro, A. Shima, and to T. Watanabe. We also classify welded string links up to homotopy, thus recovering a result of the first author with B. Audoux, P. Bellingeri and E. Wagner.