We prove some tradeoffs between the size and depth of algebraic formulae. In particular, we show that, for any fixed $ epsilon~>~O,anyalgebraicformulaofsize\s+1Ss−1canbeconvertedintoanequivalentformulaofdepth\s+1O\s-1 (log \s+1S\s-1)$ and size O(Ssup1+epsilon).
This result is an improvement over previously-known results where, to obtain the same depth bound, the formula-size is $ OMEGA (S sup alpha ),with alpha~>=~2$.