We consider algorithms for going from a "full" matrix to a condensed "band bidiagonal" form using orthogonal transformations. We use the framework of "algorithms by tiles". Within this framework, we study: (i) the tiled bidiagonalization algorithm BiDiag, which is a tiled version of the standard scalar bidiagonalization algorithm; and (ii) the R-bidiagonalization algorithm R-BiDiag, which is a tiled version of the algorithm which consists in first performing the QR factorization of the initial matrix, then performing the band-bidiagonalization of the R-factor. For both bidiagonalization algorithms BiDiag and R-BiDiag, we use four main types of reduction trees, namely FlatTS, FlatTT, Greedy, and a newly introduced auto-adaptive tree, Auto. We provide a study of critical path lengths for these tiled algorithms, which shows that (i) R-BiDiag has a shorter critical path length than BiDiag for tall and skinny matrices, and (ii) Greedy based schemes are much better than earlier proposed variants with unbounded resources. We provide experiments on a single multicore node, and on a few multicore nodes of a parallel distributed shared-memory system, to show the superiority of the new algorithms on a variety of matrix sizes, matrix shapes and core counts.