  Given a weighted graph $G=(V,E)$ with weight functions $c:E\to \mathbb{R}_+$
and $\pi:V\to \mathbb{R}_+$, and a subset $U\subseteq V$, the normalized cut
value for $U$ is defined as the sum of the weights of edges exiting $U$ divided
by the weight of vertices in $U$.
  The {\it mean isoperimetry problem}, $\mathsf{ISO}^1(G,k)$, for a weighted
graph $G$ is a generalization of the classical uniform sparsest cut problem in
which, given a parameter $k$, the objective is to find $k$ disjoint nonempty
subsets of $V$ minimizing the average normalized cut value of the parts. The
robust version of the problem seeks an optimizer where the number of vertices
that fall out of the subpartition is bounded by some given integer $0 \leq \rho
\leq |V|$.
  Our main result states that $\mathsf{ISO}^1(G,k)$, as well as its robust
version, $\mathsf{CRISO}^1(G,k,\rho)$, subjected to the condition that each
part of the subpartition induces a connected subgraph, are solvable in time
$O(k^2 \rho^2\ \pi(V(T)^3)$ on any weighted tree $T$, in which $\pi(V(T))$ is
the sum of the vertex-weights.
  This result implies that $\mathsf{ISO}^1(G,k)$ is strongly polynomial-time
solvable on weighted trees when the vertex-weights are polynomially bounded and
may be compared to the fact that the problem is NP-Hard for weighted trees in
general.
  Also, using this, we show that both mentioned problems, $\mathsf{ISO}^1(G,k)$
and $\mathsf{CRISO}^1(G,k,\rho)$ as well as the ordinary robust mean
isoperimetry problem $\mathsf{RISO}^1(G,k,\rho)$, admit polynomial-time
$O(\log^{1.5}|V| \log\log |V|)$-approximation algorithms for weighted graphs
with polynomially bounded weights, using the R{\"a}cke-Shah tree cut
sparsifier.
