Embedding complete trees into the hypercube

We consider embeddings of the complete t-ary trees of depth k (denotation T k,t) as subgraphs into the hypercube of minimum dimension n. This n, denoted by dim(T k,t), is known if max{k, t} ≤ 2. First we study the next open case max{k, t} = 3. We improve the known upper bound dim(T k,3) ≤ 2k + 1 up to limk→∞ dim(T k,3)/k ≤ 5/3 and derive the asymptotic limt→∞ dim(T 3,t)/t = 227/120. As a co-result, we present an exact formula for the dimension of arbitrary trees of depth 2, as a function of their vertex degrees. These results and new techniques provide an improvement of the known upper bound for dim(T k,t) for arbitrary k and t.