We present two new algorithms for approximating and updating the hierarchical Tucker decomposition of tensor streams. The first algorithm, Batch Hierarchical Tucker - leaf to root (BHT-l2r), proposes an alternative and more efficient way of approximating a batch of similar tensors in hierarchical Tucker format. The second algorithm, Hierarchical Tucker - Rapid Incremental Subspace Expansion (HT-RISE), updates the batch hierarchical Tucker representation of an accumulated tensor as new batches of tensors become available. The HT-RISE algorithm is suitable for the online setting and never requires full storage or reconstruction of all data while providing a solution to the incremental Tucker decomposition problem. We provide theoretical guarantees for both algorithms and demonstrate their effectiveness on physical and cyber-physical data. The proposed BHT-l2r algorithm and the batch hierarchical Tucker format offers up to $6.2\times$ compression and $3.7\times$ reduction in time over the hierarchical Tucker format. The proposed HT-RISE algorithm also offers up to $3.1\times$ compression and $3.2\times$ reduction in time over a state of the art incremental tensor train decomposition algorithm.