  Let $G$ be a simple finite connected graph with vertex set $V(G) =
\{v_1,v_2,\ldots,v_n\}$. Denote the degree of vertex $v_i$ by $d_i$ for all $1
\leq i \leq n$. The Randi\'c matrix of $G$, denoted by $R(G) = [r_{i,j}]$, is
the $n \times n$ matrix whose $(i,j)$-entry $r_{i,j}$ is $r_{i,j} =
1/\sqrt{d_id_j}$ if $v_i$ and $v_j$ are adjacent in $G$ and 0 otherwise. A tree
is a connected acyclic graph. A level-wise regular tree is a tree rooted at one
vertex $r$ or two (adjacent) vertices $r$ and $r'$ in which all vertices with
the minimum distance $i$ from $r$ or $r'$ have the same degree $m_i$ for $0
\leq i \leq h$, where $h$ is the height of $T$. In this paper, we give a
complete characterization of the eigenvalues with their multiplicity of the
Randi\'c matrix of level-wise regular trees. We prove that the eigenvalues of
the Randi\'c matrix of a level-wise regular tree are the eigenvalues of the
particular tridiagonal matrices, which are formed using the degree sequence
$(m_0,m_1,\ldots,m_{h-1})$ of level-wise regular trees.
