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In a graph $G=(V,E)$, a module is a vertex subset $M$ of $V$ such that everyvertex outside $M$ is adjacent to all or none of $M$. For example, $\emptyset$,$\{x\}$ $(x\in V )$ and $V$ are modules of $G$, called trivial modules. Agraph, all the modules of which are trivial, is prime; otherwise, it isdecomposable. A vertex $x$ of a prime graph $G$ is critical if $G - x$ isdecomposable. Moreover, a prime graph with $k$ non-critical vertices is called$(-k)$-critical graph. A prime graph $G$ is $k$-minimal if there is some$k$-vertex set $X$ of vertices such that there is no proper induced subgraph of$G$ containing $X$ is prime. From this perspective, I. Boudabbous proposes tofind the $(-k)$-critical graphs and $k$-minimal graphs for some integer $k$even in a particular case of graphs. This research paper attempts to answer I.Boudabbous's question. First, it describes the $(-k)$-critical tree. As acorollary, we determine the number of nonisomorphic $(-k)$-critical tree with$n$ vertices where $k\in \{1,2,\lfloor\frac{n}{2}\rfloor\}$. Second, it providea complete characterization of the $k$-minimal tree. As a corollary, wedetermine the number of nonisomorphic $k$-minimal tree with $n$ vertices where$k\leq 3$.

(-k)-critical trees and k-minimal trees