Invertibility of the TKF model of sequence evolution

We consider character sequences evolving on a phylogenetic tree under the TKF91 model. We show that as the sequence lengths tend to infinity the topology of the phylogenetic tree and the edge lengths are determined by any one of (a) the alignment of sequences (b) the collection of sequence lengths. We also show that the probability of any homology structure on a collection of sequences related by a TKF91 process on a tree is independent of the root location.