Dimensions of $\mathbb R$-Trees and Self-Similar Fractal Spaces of Nonpositive Curvature

We study various dimensions of spaces with nonpositive curvature in the A. D. Alexandrov sense, in particular, of $\mathbb R$-trees. We find some conditions necessary and sufficient for the metric space to be an $\mathbb R$-tree and clarify relations between the topological, Hausdorff, entropy, and rough dimensions. We build the examples of $\mathbb R$-trees and CAT(0)-spaces in which strict inequalities between the topological, Hausdorff, and entropy dimensions hold; we also show that the Hausdorff and entropy dimensions can be arbitrarily large while the topological dimension remains fixed.