Given a self-map $f$ of a compact metric space $X$, one can associate to it the induced mappings $\overline{f}$ and $\tilde{f}$ on the hyperspace $2^X$ of compact subsets of $X$ and on the hyperspace $C(X)$ of continua in $X$, respectively, both defined in a natural way. Within this framework, it is natural to investigate the relationship between individual and collective dynamics.

In this talk, we address the following question. Let $f$ be a self-map of a topological tree $T$, and let $x$ be a periodic point of $f$ with period $p$. What are the possible periods of periodic points of $\left(C(T), \tilde{f}\right)$, that is, of periodic subtrees containing $x$?

We then discuss the significance of this result for the study of further properties of the system $\left(C(T), \tilde{f}\right)$. In particular, using this result, we show that the induced system is always almost equicontinuous and we characterize its Birkhoff center.

\emph{The talk is based on a joint work with Piotr Oprocha.}