Persistent homology with non-contractible preimages

For a fixed $N$, we analyze the space of all sequences $z=(z_1,\dots,z_N)$, approximating continuous function on circle, with given persistence diagram $P$, and show that typical components this are homotopy equivalent to $S^1$. We also consider functions $Y$-shaped (resp., star-shaped) trees 2-point diagram, is $S^1$ bouquet circles).