We present an alternative layer to convolution layers in convolutional neural networks (CNNs). Our approach reduces the complexity of convolutions by replacing it with binary decisions. 
Those binary decisions are used as indexes to conditional probability distributions where each probability represents a leaf in a decision tree. 
This means that only the indices to the probabilities need to be determined once, thus reducing the complexity of convolutions by the depth of the output tensor. 
Index computation is performed by simple binary decisions that require fewer CPU cycles compared to conventionally used multiplications. 
In addition, we show how convolutions can be replaced by binary decisions. 
These binary decisions form indices in the conditional probability distributions and we show how they are used to replace 2D weight matrices as well as 3D weight tensors. 
These new layers can be trained like convolution layers in CNNs based on the backpropagation algorithm, for which we provide a formalization. 
Our results on multiple publicly available data sets show that our approach outperforms conventional CNNs. 
Beyond the formalized reduction of complexity and the improved qualitative performance, we show empirically a significant runtime improvement compared to convolution layers.