Abstract    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.
['Jeremy Gibbons']
Computing Downwards Accumulations on Trees Quickly