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_h of height h into the square n×n grid we consider the layout´s expansion ratio r=n²/(2^h+1 -1). For r the least expansion among layouts of T_h into grids, we obtain the bounds 1.122⩽r⩽1.4656 for large h and n. For layouts into extended grids we similarly obtain bounds 1.03⩽r⩽1.115. The upper bounds are improvements on previous work, while the lower bounds are the first such to appear