  Twenty years ago, Werner Krandick defined two statistics on binary trees. The
first one determines the number of jumps, when traversing the tree in
depth-first-search, from a vertex to one closer to the root, and the second
keeps tracks of the sum of the jump-distances. He used clever but ad hoc
human-generated arguments to find explicit expressions for their expectations.
In this methodological note, we illustrate the power of experimental
mathematics and symbolic computation to do much more. We derive closed-form
expressions for the actual weight-enumerators according to these statistics
(from which not only the expectations, but also the variances, and as many
higher moments as desired, can be obtained). We also actually give the first
eight moments, and conjecture that the first statistic (number of jumps) is
asymptotically normal, and prove that the second one (sum of jump distances) is
definitely not.
