  Let $G$ be a graph and let $f$ be a positive integer-valued function on
$V(G)$ satisfying $2m\le f\le b$, where $b$ and $m$ are two positive integers
with $b\ge 4m^2$. In this paper, we show that if $G$ is $b^2$-tough and
$|V(G)|\ge b^2$, then it has an $m$-tree-connected factor $H$ such that for
each vertex $v$, $$d_H(v)\in \{f(v), f(v)+1\}.$$ Next, we generalize this
result by giving sufficient conditions for a tough graph to have a
tree-connected factors $H$ such that for each vertex $v$, $d_H(v)\in \{f(v),
f(v)+k\}$. As an application, we prove that every $64b(b-a)^2$-tough graph $G$
of order at least $b+1$ with $ab|V(G)|$ even admits a connected factor whose
degrees lie in the set $\{a,b\}$, where $a$ and $b$ are two integers with $2\le
a< b < \frac{6}{5}a$. Moreover, we prove that every $16$-tough graph $G$ of
order at least three admits a $2$-connected factor whose degrees lie in the set
$\{2,3\}$, provided that $G$ has a $2$-factor with girth at least five. This
result confirms a weaker version of a long-standing conjecture due to Chv\'atal
(1973).
