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Let $G_n=\mathrm{GL}_n(F)$ be the general linear group over a non-Archimedeanlocal field $F$. We formulate and prove a necessary and sufficient condition ondetermining when \[ \mathrm{Hom}_{G_n}(\pi, \pi') \neq 0 \] for irreduciblesmooth representations $\pi$ and $\pi'$ of $G_{n+1}$ and $G_n$ respectively.This resolves the problem of the quotient branching law.  We also prove that any simple quotient of a Bernstein-Zelevinsky derivativeof an irreducible representation can be constructed by a sequence ofderivatives of essentially square-integrable representations. This resulttransferred to affine Hecke algebras of type A gives a generalization of theclassical Pieri's rule of symmetric groups.  One key new ingredient is a characterization of the layer in theBernstein-Zelevinsky filtration that contributes to the branching law, obtainedby the multiplicity one theorem for standard representations, which also givesa refinement of the branching law. Another key new ingredient is constructionsof some branching laws and simple quotients of Bernstein-Zelevinsky derivativesby taking certain highest derivatives.

Quotient branching law for $p$-adic $(\mathrm{GL}_{n+1}, \mathrm{GL}_n)$ I: generalized Gan-Gross-Prasad relevant pairs