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Diamond graphs and binary trees are important examples in the theory ofmetric embeddings and also in the theory of metric characterizations of Banachspaces. Some results for these families of graphs are parallel to each other,for example superreflexivity of Banach spaces can be characterized both interms of binary trees (Bourgain, 1986) and diamond graphs (Johnson-Schechtman,2009). In this connection, it is natural to ask whether one of these familiesadmits uniformly bilipschitz embeddings into the other. This question wasanswered in the negative by Ostrovskii (2014), who left it open to determinethe order of growth of the distortions. The main purpose of this paper is toget a sharp-up-to-a-logarithmic-factor estimate for the distortions ofembeddings of binary trees into diamond graphs, and, more generally, intodiamond graphs of any finite branching $k\ge 2$. Estimates for distortions ofembeddings of diamonds into infinitely branching diamonds are also obtained.

Distortion of embeddings of binary trees into diamond graphs