  Let $\mathcal{X}$ be a Banach space over the complex field $\mathbb{C}$ and
$\mathcal{B(X)}$ be the algebra of all bounded linear operators on
$\mathcal{X}$. Let $\mathcal{N}$ be a non-trivial nest on $\mathcal{X}$, ${\rm
Alg}\mathcal{N}$ be the nest algebra associated with $\mathcal{N}$, and
$L\colon {\rm Alg}\mathcal{N}\longrightarrow \mathcal{B(X)}$ be a linear
mapping. Suppose that $p_n(x_1,x_2,\cdots,x_n)$ is an $(n-1)$-th commutator
defined by $n$ indeterminates $x_1, x_2, \cdots, x_n$. It is shown that $L$
satisfies the rule $$ L(p_n(A_1, A_2, \cdots, A_n))=\sum_{k=1}^{n}p_n(A_1,
\cdots, A_{k-1}, L(A_k), A_{k+1}, \cdots, A_n) $$ for all $A_1, A_2, \cdots,
A_n\in {\rm Alg}\mathcal{N}$ if and only if there exist a linear derivation
$D\colon {\rm Alg}\mathcal{N}\longrightarrow \mathcal{B(X)}$ and a linear
mapping $H\colon {\rm Alg}\mathcal{N}\longrightarrow \mathbb{C}I$ vanishing on
each $(n-1)$-th commutator $p_n(A_1,A_2,\cdots, A_n)$ for all $A_1, A_2,
\cdots, A_n\in {\rm Alg}\mathcal{N}$ such that $L(A)=D(A)+H(A)$ for all $A\in
{\rm Alg}\mathcal{N}$.
