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In this paper, we focus on the prediction phase of a random forest and studythe problem of representing a bag of decision trees using a smaller bag ofdecision trees, where we only consider binary decision problems on the binarydomain and simple decision trees in which an internal node is limited toquerying the Boolean value of a single variable. As a main result, we show thatthe majority function of $n$ variables can be represented by a bag of $T$ ($<n$) decision trees each with polynomial size if $n-T$ is a constant, where $n$and $T$ must be odd (in order to avoid the tie break). We also show that a bagof $n$ decision trees can be represented by a bag of $T$ decision trees eachwith polynomial size if $n-T$ is a constant and a small classification error isallowed. A related result on the $k$-out-of-$n$ functions is presented too.

On the Trade-off between the Number of Nodes and the Number of Trees in a Random Forest