Efficient Embeddings of Binary Trees in VLSI Arrays

We consider the problem of embedding a complete binary tree in square- or hexagonally-connected VLSI arrays of processing elements (PE's). This problem can be solved in a radically different manner from current layout techniques which are aimed at laying out a given graph in the plane. The difference is due to the fact that a PE can be used both as a tree node and as a connecting element between distant nodes. New embedding schemes are presented in which (asymptotically) 100 percent of the PE's are utilized as tree nodes. This is a significant savings over known schemes, which achieve 50 percent utilization (the well-known H-tree) and 71 percent for some hexagonal schemes. These schemes also speed up signal propagation from the root to the leaves.