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We prove that a Banach space X is not super-reflexive if and only if thehyperbolic infinite tree embeds metrically into X. We improve one implicationof J.Bourgain's result who gave a metrical characterization ofsuper-reflexivity in Banach spaces in terms of uniforms embeddings of thefinite trees. A characterization of the linear type for Banach spaces is givenusing the embedding of the infinite tree equipped with a suitable metric.

Metrical characterization of super-reflexivity and linear type of Banach spaces