State-of-the-art branch and bound algorithms for mixed integer programming make use of special methods for making branching decisions. Strategies that have gained prominence include modern variants of so-called strong branching (Applegate, et al.,1995) and reliability branching (Achterberg, Koch and Martin, 2005; Hendel, 2015), which select variables for branching by solving associated linear programs and exploit pseudo-costs (Benichou et al., 1971). We suggest new branching criteria and propose alternative branching approaches called narrow gauge and analytical branching. The perspective underlying our approaches is to focus on prioritization of child nodes to examine fewer candidate variables at the current node of the B&B tree, balanced with procedures to extrapolate the implications of choosing these candidates by generating a small-depth look-ahead tree. Our procedures can also be used in rules to select among open tree nodes (those whose child nodes have not yet been generated). We incorporate pre- and post-winnowing procedures to progressively isolate preferred branching candidates, and employ derivative (created) variables whose branches are able to explore the solution space more deeply.