  Predicting the nodes of a given graph is a fascinating theoretical problem
with applications in several domains. Since graph sparsification via spanning
trees retains enough information while making the task much easier, trees are
an important special case of this problem. Although it is known how to predict
the nodes of an unweighted tree in a nearly optimal way, in the weighted case a
fully satisfactory algorithm is not available yet. We fill this hole and
introduce an efficient node predictor, Shazoo, which is nearly optimal on any
weighted tree. Moreover, we show that Shazoo can be viewed as a common
nontrivial generalization of both previous approaches for unweighted trees and
weighted lines. Experiments on real-world datasets confirm that Shazoo performs
well in that it fully exploits the structure of the input tree, and gets very
close to (and sometimes better than) less scalable energy minimization methods.
