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Among those real symmetric matrices whose graph is a given tree $T$, themaximum multiplicity $M(T)$ that can be attained by an eigenvalue is known tobe the path cover number of $T$. We say that a tree is $k$-NIM if, whenever aneigenvalue attains a multiplicity of $k-1$ less than the maximum multiplicity,all other multiplicities are $1$. $1$-NIM trees are known as NIM trees, and acharacterization for NIM trees is already known. Here we provide agraph-theoretic characterization for $k$-NIM trees for each $k\geq 1$, as wellas count them. It follows from the characterization that $k$-NIM trees exist on$n$ vertices only when $k=1,2,3$. In case $k=3$, the only $3$-NIM trees aresimple stars.

$k$-NIM trees: Characterization and Enumeration