  For $k\geq 0$, a $k$-generalized Markov number is an integer which appears in
some positive integer solution to the $k$-generalized Markov equation $x^2 +
y^2 + z^2 + k(yz + zx + xy) = (3 + 3k)xyz$. In this paper, we discuss a
combinatorial structure of generalized Markov numbers. To investigate this
structure in detail, we use two families of matrices: the $k$-generalized Cohn
matrices and the $k$-Markov-monodromy matrices, which are elements of $SL(2,
\mathbb{Z})$ whose $(1,2)$-entries are $k$-generalized Markov numbers. We show
that these two families of matrices recover the tree structure of the positive
integer solutions to the generalized Markov equation, and we give geometric
interpretations and a combinatorial interpretation of $k$-generalized Markov
numbers. As an application, we provide a computation algorithm of classical
Markov number from a one-dimensional dynamical viewpoint. Moreover, we clarify
a relation between $k$-generalized Markov numbers and toric surface
singularities via continued fractions.
