On mapping complete binary trees into grids and extended grids

We consider one-to-one, congestion one layouts of complete binary trees into 2-D grids and extended 2-D grids. For such a layout of the complete binary tree T/sub h/ of height h into the square n/spl times/n grid we consider the layout's expansion ratio r=n/sup 2//(2/sup h+1/-1). For r the least expansion among layouts of T/sub h/ into grids, we obtain the bounds 1.122/spl les/r/spl les/1.4656 for large h and n. For layouts into extended grids we similarly obtain bounds 1.03/spl les/r/spl les/1.115. The upper bounds are improvements on previous work, while the lower bounds are the first such to appear.