Computing downwards accumulations on trees quickly

Downwards passes on binary trees are essentially functions which pass information down a tree, from the root towards the leaves. Under certain conditions, a downwards pass is both 'efficient' (computable in a functional style in parallel time proportional to the depth of the tree) and 'manipulable' (enjoying a number of distributivity properties useful in program construction); we call a downwards pass satisfying these conditions a downwards accumulation. In this paper, we show that these conditions do in fact yield a stronger conclusion: the accumulation can be computed in parallel time proportional to the logarithm of the depth of the tree, on a CREW PRAM machine