On Embedding Binary Trees into Hypercubes

Hypercubes are known to be able to simulate other structures such as grids and binary trees. It has been shown that an arbitrary binary tree can be embedded into a hypercube with constant expansion and constant dilation. This paper presents a simple linear-time heuristic which embeds an arbitrary binary tree into a hypercube with expansion 1 and average dilation no more than 2. We also give some results extending good embeddings for parity-balanced binary trees to arbitrary binary trees. In particular, we show that a conjecture of Havel [8] implies embeddings of binary trees into hypercubes with expansion 1 and either dilation 2 or average dilation approaching 1, and embeddings with expansion 2 and dilation 1.