Expansion of layouts of complete binary trees into grids

Let Th be the complete binary tree of height h. Let M be the infinite grid graph with vertexset Z, where two vertices (x1,y1) and (x2,y2) of M are adjacent if and only if|x1-x2| + |y1-y2| = 1. Suppose that T is a tree which is a subdivision of Th and is also isomorphicto a subgraph of M. Motivated by issues in optimal VLSI design, we show that the point expansionration(T)n(Th)=n(T)2 1is bounded below by 1.122 for h sufficiently large. That is, we give bounds on how many vertices of degree 2 must be inserted along the edges of Th in order that the resultingtree can be laid out in the grid.Concerning the constructive end of VLSI design, suppose that T is a tree which is asubdivision of Th and is also isomorphic to a subgraph of the n × n grid graph. Define theexpansion ratio of such a layout to benn(Th)=n2-1 . We show constructively that the minimumpossible expansion ratio over all layouts of Th is bounded above by 1.4656 for sufficiently large h.That is, we give efficient layouts of complete binary trees into square grids, making improvementsupon the previous work of others.We also give bounds for the point expansion and expansion problems for layouts of Th intoextended grids, i.e. grids with added diagonals.