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Let $P$ be a polynomial of degree $d$ with Julia set $J_P$. Let $\widetildeN$ be the number of non-repelling cycles of $P$. By the famous Fatou-Shishikurainequality $\widetilde N\le d-1$. The goal of the paper is to improve thisbound. The new count includes \emph{wandering collections of non-precriticalbranch continua}, i.e., collections of continua or points $Q_i\subset J_P$\emph{all} of whose images are pairwise disjoint, contain no critical points,and contain the limit sets of $\mathrm{eval}(Q_i)\ge 3$ external rays. Also, werelate individual cycles, which are either non-repelling or repelling with noperiodic rays landing, to individual critical points that are recurrent in aweak sense.  A weak version of the inequality reads \[ \widetilde N+N_{irr}+\chi+\sum_i(\mathrm{eval}(Q_i)-2) \le d-1 \] where $N_{irr}$ counts repelling cycles withno periodic rays landing at points in the cycle, $\{Q_i\}$ form a wanderingcollection $\mathcal{B}_\mathbb{C}$ of non-precritical branch continua,$\chi=1$ if $\mathcal{B}_\mathbb{C}$ is non-empty, and $\chi=0$ otherwise.

An Extended Fatou-Shishikura inequality and wandering branch continua for polynomials