  The aim of this paper is to provide upper bounds for the entropy numbers of
summation operators on trees in a critical case. In a recent paper [10] we
elaborated a framework of weighted summation operators on general trees where
we related the entropy of the operator with those of the underlying tree
equipped with an appropriate metric. However, the results were left incomplete
in a critical case of the entropy behavior, because this case requires much
more involved techniques. In the present article we fill the gap left open in
[10]. To this end we develop a method, working in the context of general trees
and general weighted summation operators, which was recently proposed in [9]
for a particular critical operator on the binary tree. Those problems appeared
in natural way during the study of compactness properties of certain Volterra
integral operators in a critical case.
