In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest. A well-developed example is the investigation of how the parameter treewidth influences the complexity of (other) graph problems. The reason why such investigations are of general interest is that real-world input distributions for computational problems often inherit structure from the natural computational processes that produce the problem instances (not necessarily in obvious, or well-understood ways). The max leaf number ml(g) of a connected graph g is the maximum number of leaves in a spanning tree for g. Exploring questions analogous to the well-studied case of treewidth, we can ask: how hard is it to solve 3-coloring, hamilton path, minimum dominating set, minimum bandwidth or many other problems, for graphs of bounded max leaf number? what optimization problems are w-hard under this parameterization? we do two things: (1) we describe much improved fpt algorithms for a large number of graph problems, for input graphs g for which ml(g)=k, based on the polynomial-time extremal structure theory canonically associated to this parameter. We consider improved algorithms both from the point of view of kernelization bounds, and in terms of improved fixed-parameter tractable (fpt) runtimes o *(f(k)). (2) the way that we obtain these concrete algorithmic results is general and systematic. We describe the approach, and raise programmatic questions.