  We show that the asymptotic regularity and the strong convergence of the
modified Halpern iteration due to T.-H. Kim and H.-K. Xu and studied further by
A. Cuntavenapit and B. Panyanak and the Tikhonov-Mann iteration introduced by
H. Cheval and L. Leu\c{s}tean as a generalization of an iteration due to Y. Yao
et al. that has recently been studied by Bo\c{t} et al. can be reduced to each
other in general geodesic settings. This, in particular, gives a new proof of
the convergence result in Bo\c{t} et al. together with a generalization from
Hilbert to CAT(0) spaces. Moreover, quantitative rates of asymptotic regularity
and metastability due to K. Schade and U. Kohlenbach can be adapted and
transformed into rates for the Tikhonov-Mann iteration corresponding to recent
quantitative results on the latter of H. Cheval, L. Leu\c{s}tean and B. Dinis,
P. Pinto respectively. A transformation in the converse direction is also
possible. We also obtain rates of asymptotic regularity of order $O(1/n)$ for
both the modified Halpern (and so in particular for the Halpern iteration) and
the Tikhonov-Mann iteration in a general geodesic setting for a special choice
of scalars.
