Minimum congestion embedding of complete binary trees into tori

We consider the problem of embedding complete binary trees into 2-dimensional tori with minimum (edge) congestion. The embedding proposed here achieves the minimum congestion and an almost minimum size of a torus (up to the constant term of 1). In particular, the embedding is optimal for the problem of embedding a 2n-1-vertex complete binary tree with an even integer n into a square torus with unit congestion.