Tensor networks developed in the context of condensed matter physics try to approximate order-$N$ tensors with a reduced number of degrees of freedom that is only polynomial in $N$ and arranged as a network of partially contracted smaller tensors. As we have recently demonstrated in the context of quantum many-body physics, computation costs can be further substantially reduced by imposing constraints on the canonical polyadic (CP) rank of the tensors in such networks [arXiv:2205.15296]. Here, we demonstrate how tree tensor networks (TTN) with CP rank constraints and tensor dropout can be used in machine learning. The approach is found to outperform other tensor-network-based methods in Fashion-MNIST image classification. A low-rank TTN classifier with branching ratio $b=4$ reaches a test set accuracy of 90.3\% with low computation costs. Consisting of mostly linear elements, tensor network classifiers avoid the vanishing gradient problem of deep neural networks. The CP rank constraints have additional advantages: The number of parameters can be decreased and tuned more freely to control overfitting, improve generalization properties, and reduce computation costs. They allow us to employ trees with large branching ratios, substantially improving the representation power.