  A harmonic cycle $\lambda$, also called a discrete harmonic form, is a
solution of the Laplace's equation with the combinatorial Laplace operator
obtained from the boundary operators of a chain complex. By the combinatorial
Hodge theory, harmonic spaces are isomorphic to the homology groups with real
coefficients. In particular, if a cell complex has a one dimensional reduced
homology, it has a unique harmonic cycle up to scalar, which we call the
\emph{standard harmonic cycle}. In this paper, we will present a formula for
the standard harmonic cycle $\lambda$ of a cell complex based on a
high-dimensional generalization of cycletrees. Moreover, by using duality, we
will define the standard harmonic cocycle $\lambda^*$, and show intriguing
combinatorial properties of $\lambda$ and $\lambda^*$ in relation to (dual)
spanning trees, (dual) cycletrees, winding numbers $w(\cdot)$ and cutting
numbers $c(\cdot)$ in high dimensions.
